literacy strategies for improving mathematics instruction

Education$21.95 US

What makes mathematics so confusing to students? To succeed in the studyof arithmetic, geometry, or algebra, students must learn what is effectively asecond language of mathematical terms and symbols. In Literacy Strategies forImproving Mathematics Instruction, Joan M. Kenney and her coauthors describecommon ways in which students misinterpret the language of mathematics,and show teachers what they can do to ensure that their students become flu-ent in that language.

The authors synthesize the research on what it takes to decode mathematicaltext, explain how teachers can use guided discourse and graphic representa-tions to help students develop mathematical literacy skills, offer guidance onusing action research to enhance mathematics instruction, and discuss theimportance of student-centered learning and concept-building skills in theclassroom. Real-life vignettes of student struggles illuminate the profoundeffect of literacy problems on student achievement in mathematics.

This book will help teachers better understand their students’ difficulties withmathematics and take the steps necessary to alleviate them. Abundantlyresearched and filled with helpful strategies and resources, it is an invaluableresource for mathematics teachers at all levels.

Joan M. Kenney has been both a research scientist and a mathematics teacherat the secondary and college levels. Most recently, she served as codirector ofthe Balanced Assessment Program at the Harvard University Graduate Schoolof Education.

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Library of Congress Cataloging-in-Publication Data

Literacy strategies for improving mathematics instruction / Joan M. Kenney . . . [et al.].p. cm.

Includes bibliographical references and index.ISBN 1-4166-0230-5 (pbk. : alk. paper) 1. Mathematics—Study and teaching.

I. Kenney, Joan M., 1937– II. Association for Supervision and Curriculum Development.

QA11.2.L58 2005510�.71—dc22 2005016434

12 11 10 09 08 07 06 05 12 11 10 9 8 7 6 5 4 3 2 1

®

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Preface ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ v

Chapter 1: Mathematics as Language ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1Joan M. Kenney

Chapter 2: Reading in the Mathematics Classroom~ ~ ~ ~ 9Diana Metsisto

Chapter 3: Writing in the Mathematics Classroom ~ ~ ~ 24Cynthia L. Tuttle

Chapter 4: Graphic Representation in the Mathematics Classroom~ ~ ~ ~ ~ ~ ~ ~ 51Loretta Heuer

Chapter 5: Discourse in the Mathematics Classroom ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 72Euthecia Hancewicz

Chapter 6: Creating Mathematical Metis ~ ~ ~ ~ ~ ~ ~ ~ 87Joan M. Kenney

Appendix: Structured Agendas Used to Research This Book ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 95

References and Resources ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 99

Index ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 105

About the Authors ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 111

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v

School days, school days,Dear old Golden Rule days.Reading and ‘Riting and ‘Rithmetic,Taught to the tune of a hick-ry stick . . .

At the time this song was popular, the mark of an educated person was basicfacility in each of the disciplines of reading, writing, and mathematics. Thesewere viewed as discrete topics and taught in relative isolation of each other,although effective writing was seen as a somewhat logical result of skill inreading. The notion that mathematics textbooks were to be read for content,or that mathematics problems were to be solved by producing anything otherthan a numerical answer, was totally foreign. Indeed, reading and writing inmathematics did not become a subject of serious discussion and research untilthe appearance of the standards-based mathematics curricula of the 1990s,driven by the National Council of Teachers of Mathematics (NCTM) stan-dards of 1991. Suddenly a clearer distinction between arithmetic and mathe-matics was put forward. The ability to master and demonstrate mathematicalknowledge came to be seen as the result of a process that involves teachingfor understanding, student-centered learning, concept-building rather thanmemorization of facts, and the ability to communicate mathematical under-standing to others.

This book blends current research on selected aspects of language literacywith practitioner evidence of the unique challenges presented in transferringthese language skills to the mathematics classroom. It is distinctive in that it

Preface

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vi Literacy Strategies for Improving Mathematics Instruction

not only looks extensively at current literacy research, but also connects theresearch to actual practice and the development of concrete classroom strate-gies. We present the etymological underpinnings for thinking about mathemat-ics as a language, and describe the unique difficulties encountered in readingmathematics text and creating a mathematics vocabulary. We also look at thenew role of mathematics teacher as reading teacher, and differentiate betweenwhat literacy looks like in the language arts classroom and in the mathematicsclassroom. Most importantly, however, this book is designed to provide educa-tors with a broad spectrum of tools to use as they work to transform studentsinto confident communicators and practitioners of mathematics.

Joan M. KenneyBalanced Assessment Program

Euthecia HancewiczDiana Metsisto Loretta HeuerCynthia L. TuttleMassachusetts Mathematics Coaching Project

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1

1

Over the years, we have colloquially referred to mathematics as a special lan-guage. Yet practitioners have had little research to consult on the matter, orimpetus to reflect on whether the process of learning the language of mathe-matics is similar to that of learning any other second language.

Although researchers have paid lip service to the unique vocabulary ofmathematics, they have done little to highlight the ambiguities, doublemeanings, and other “word” problems associated with the discipline. Igno-rance of these issues can lead to impaired communication at best, and seriousmathematical misunderstanding at worst. To compound the difficulty, infor-mation in mathematics texts is presented in a bewildering assortment ofways; in attempting to engage students, textbook writers too often introducegraphic distraction, and format the pages in ways that obscure the basic con-cepts. In this book we look carefully and reflectively at the difficulties inher-ent in learning the language of mathematics, and suggest strategies for howbest to overcome them.

There are over 4,000 languages and dialects in the world, and all of themshare one thing in common: they have a category for words representingnouns, or objects, and a category for words representing verbs, or actions.Taking this commonality as a starting point provides an interesting way oflooking at the mathematical world and its language. It is possible to identifyboth content and process dimensions in mathematics, but unlike many dis-ciplines, in which process refers to general reasoning and logic skills, in math-ematics the term refers to skills that are domain-specific. As a result, peopletend to lump content and process together when discussing mathematics,calling it all mathematics content. However, it is vitally important to maintaina distinction between mathematical content and process, because the distinc-tion reflects something very significant about the way humans approach

Mathematics as LanguageJoan M. Kenney

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2 Literacy Strategies for Improving Mathematics Instruction

mental activity of any sort. All human languages have grammatical structuresthat distinguish between nouns and verbs; these structures express the dis-tinction between the objects themselves and the actions carried out by or onthe objects.

A model first proposed by the Balanced Assessment Program at the HarvardGraduate School of Education (Schwartz & Kenney, 1995) suggests that wethink about mathematical nouns, or objects, as being numbers, measurements,shapes, spaces, functions, patterns, data, and arrangements—items that com-fortably map onto commonly accepted mathematics content strands. Mathe-matical verbs may be regarded as the four predominant actions that we ascribeto problem-solving and reasoning:

• Modeling and formulating. Creating appropriate representations and rela-tionships to mathematize the original problem.

• Transforming and manipulating. Changing the mathematical form inwhich a problem is originally expressed to equivalent forms that representsolutions.

• Inferring. Applying derived results to the original problem situation, andinterpreting and generalizing the results in that light.

• Communicating. Reporting what has been learned about a problem to aspecified audience.

Taken as a whole, these four actions represent the process that we go throughto solve a problem. Taken individually, they represent actions that students candevelop and on which they can be assessed. To view the actions individuallyalso enables us to separate the type of proficiency required in, for instance,manipulation and transformation, which are primarily skill-based actions, fromthe more complex proficiencies required to create a mathematical model andto generalize and extend the results of a mathematical action. Also, because notall exercises make equal demands on or involve equivalent competency in eachof the mathematical objects and actions, students will not necessarily performevenly across them. For example, the ability of learners to model a probleminvolving functions may be quite different from their ability to model a prob-lem involving data. Students may vary in their abilities to communicate theirunderstanding of geometric objects, and objects of number and quantity. Hereagain, it is important to keep the distinction between mathematical objects andactions explicit while viewing student work; otherwise, comprehensive assess-ment of student mathematical understanding will be severely limited.

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3Mathematics as Language

According to Schwartz (1996), dividing the elements of mathematics intoobjects and actions has significant implications for curriculum:

To a large extent the arithmetic curriculum of the elementary school aswell as the algebra curriculum of the middle and high school focus on themanipulation of symbols representing mathematical objects, rather than onusing mathematical objects in the building and analyzing of arithmetic oralgebraic models. Thus, in the primary levels, most of the mathematicaltime and attention of both teachers and students is devoted to the teach-ing and learning of the computational algorithms for the addition, sub-traction, multiplication, and division of integers and decimal andnon-decimal fractions. Later, the teaching and learning of algebrabecomes, in large measure, the teaching and learning of the algebraic nota-tional system and its formal, symbolic manipulation. Using the mathemat-ical objects and actions as the basis for modeling one’s surround is aneglected piece of the mathematics education enterprise. (p. 39)

But before we can use mathematical objects to model our surround, wemust first acquire them. For many reasons, this is an extremely difficult pro-cess. Mathematics truly is a foreign language for most students: it is learnedalmost entirely at school and is not spoken at home. Mathematics is not a“first” language; that is, it does not originate as a spoken language, except forthe naming of small whole numbers. Mathematics has both formal and infor-mal expressions, which we might characterize as “school math” and “streetmath” (Usiskin, 1996). When we attempt to engage students by using real-world examples, we often find that the colloquial or “street” language doesnot always map directly or correctly onto the mathematical syntax. For exam-ple, suppose a pre-algebra student is asked to symbolically express that thereare twice as many dogs as cats in the local animal shelter. The equation 2C=D describes the distribution, but is it true that two cats are equivalent toone dog?

Recasting the mathematics domain into objects and actions can also help toilluminate the similarities and differences between how we learn the languageof mathematics and how we learn any other second language. My coauthorsand I were all viewed as “good in math”—that is, fluent in the mathematicslanguage. As we compared our earliest memories of learning mathematics, oneof us remembered being made aware of numbers as an abstract quantity bylooking at the pattern of classroom windowpanes. Another, who characterizes

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herself as a very visual learner, recalled seeing boxes of six pansies apiecepacked in cartons of four boxes apiece at her family’s floral business—an obser-vation that triggered previously undiscovered ideas about addition and multi-plication. Another member of the group, whose father was a banker,remembered dinner-table conversations filled with mental mathematics prob-lems, yet she tends to rely heavily on writing as a learning tool. And one of ustold of feeling suddenly “divorced” from the language of mathematics uponentering Algebra II, where everything became symbol-laden and obscure.

Despite our varying introductions to and degrees of comfort with mathe-matical language, my coauthors and I have retained knowledge of the lan-guage of mathematics far better than knowledge of the spoken languages westudied. Even though most of us took two or three years of a foreign languagein high school, we have not been able to sustain our use of it in any expan-sive way. Certainly some of this can be explained by our lack of daily use ofthe languages, but another factor may be at work—namely, the way welearned them, mainly by memorizing vocabulary words and verb conjuga-tions out of any immediate context. Is this not similar to the way studentsmay successfully memorize number facts and plug into algorithms whenlearning arithmetic? However, when these students are later asked to drawinferences, discriminate between quantities, or justify solutions, the full effectof their lack of mathematics fluency becomes apparent.

Another interesting commonality between mathematics and foreign lan-guages lies in the relationship between rhyme and retention. Several in ourgroup either did not initially speak English as a first language, or were essen-tially bilingual for a period of time in both a “home” language and a “school”language. What we tend to retain of the second language is most easilyaccessed through music or rhyme—we remember songs, prayers, and poemseven though we can no longer perform even the most rudimentary task ofwritten or spoken communication. This brings to mind students who are ableto spout mathematical facts using jingles or mnemonics, but cannot use thefacts in any extended way or for any new purpose.

Perhaps the greatest difficulty in learning the language of mathematics isthat a double decoding must go on during the entire process. Particularly in theearly stages, we must decode spoken mathematics words in the initial contextof normal parlance, and then translate to the different context of mathematicsusage. Double decoding also occurs when we first encounter written mathe-matics words or symbols, which must first be decoded, and then connected to

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a concept that may or may not be present in prior knowledge even in an ele-mentary way.

As developmentally complex as double decoding is for most students,imagine how difficult it must be for second-language learners. The followinganecdote illustrates the problem.

As I observed a 4th grade classroom, the teacher began by discussing wholenumbers; she then moved to the distinction between even and odd numbers.When asked to classify numbers as even or odd, one of the students, a recentHispanic immigrant with limited English skills, consistently marked the num-bers 6 and 10 as odd. When asked to explain, he said, “Those whole numbersare not multiples of 2.” Additional conversation between teacher and studentdid little to clarify the problem until I asked the student what he meant by“whole numbers.” It was only when he answered “6, 8, 9, 10, and maybe 3”that we realized that this student had constructed a mental model of “hole”numbers—that is, numbers formed by sticks and holes—and that this was, tohim, a totally consistent explanation. If a number had only one “hole,” likethe numbers 6, 9, and 10, it was odd, because the number of “holes” was nota multiple of 2. The number 3 was problematic in this student’s system,because he couldn’t decide whether it was really two holes if you completedthe image, or two half-holes that could be combined to make a single one.Though this anecdote may seem bizarre, it richly illustrates the difficulty stu-dents have as they struggle to make meaning of the words they hear in themathematics classroom.

Another difficulty inherent in the decoding process stems from the factthat, although most mathematical nouns actually describe the things theyrefer to, their origins are usually Latin or Greek rather than English. The workof Steven Schwartzman (1994) has traced these connections. The mathemat-ics words that we use in English come from many sources, and have assumedtheir current forms as a result of various processes; in addition, many containmore than one unit of meaning. Even though English, Greek, and Latin areall rooted in the Indo-European language—the common ancestor of the lan-guages spoken by roughly half of the people in the world today—few Ameri-can students currently have any exposure to either Greek or Latin.

An interesting example of how language can either illuminate or obscureconcepts is the difference between the word “twelve” in English and the corre-sponding word in Chinese, the grammar of which is a perfect reflection ofdecimal structure. One day, as I was observing the piloting of a manipulative

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device designed to help students understand place value, I talked at lengthwith a Chinese-born teacher who was using the device in his classroom. Hetold me that in his native language there are only nine names for the num-bers 1 through 9, and three multipliers (10, 100, and 1,000). In order to namea number, you read its decomposition in base 10, so that 12 means “ten andtwo.” This elegant formalism contrasts sharply with the 29 words needed toexpress the same numbers in English, where, in addition to the words for thenumbers 1 through 9, there are special words for the numbers 11 through 19and the decades from 20 to 90, none of which can be predicted from thewords for the other numerals. To compound the confusion, the English wordfor 12 incorporates two units of meaning: The first part of the word comesfrom Latin and Greek expressions for “two,” and the second part is related toan Indo-European root meaning “leave” (Wylde & Partridge, 1963). Thus, anetymological decoding would be “the number that leaves 2 behind when 10,the base in which we do our calculating, is subtracted from it”—far fromtransparent to the novice learner!

It is also important to recognize the potential for enormous confusion thatsymbolic representations can create. As Barton and Heidema (2002) note:

In reading mathematics text one must decode and comprehend not onlywords, but also signs and symbols, which involve different skills. Decodingwords entails connecting sounds to the alphabetic symbols, or letters. . . .In contrast, mathematics signs and symbols may be pictorial, or they mayrefer to an operation, or to an expression. Consequently, students need tolearn the meaning of each symbol much like they learn “sight” words inthe English language. In addition they need to connect each symbol, theidea it represents, and the written or spoken term that corresponds to theidea. (p. 15)

The confounding potential of symbolic representation cannot be over-stated. Younger students can be quite mystified by the fact that changing theorientation of a symbol—for example, from horizontal (=) to vertical (||)—cancompletely change its meaning. Figure 1.1 is a collection of confusing words,symbols, and formats that we have encountered in the classroom. This is notan exhaustive list; rather, it is intended as a work-in-progress that teachers areencouraged to add to, and as an early-warning system for educators who aremystified by the misinterpretations particular students may attribute to a math-ematical situation that, to others, has quite a different meaning.

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FIGURE 1.1Confusing Terms, Formats, and Symbols in Mathematics

� and )

• , � , ( ) , and *÷ , ) , /, and

m__n

� , � , � , � , and �< and >


To summarize, in mathematics, vocabulary may be confusing because thewords mean different things in mathematics and nonmathematics contexts,because two different words sound the same, or because more than one wordis used to describe the same concept. Symbols may be confusing either becausethey look alike (e.g., the division and square root symbols) or because differ-ent representations may be used to describe the same process (e.g., •, *, and� for multiplication). Graphic representations may be confusing because of for-matting variations (e.g., bar graphs versus line graphs) or because the graph-ics are not consistently read in the same direction.

Throughout this book, we will explore how mathematics instruction canbe made deeper and more stimulating through skill-building in reading andwriting. We will also discuss the importance of graphic representations and classroom discourse. As Barnett-Clarke and Ramirez (2004) note: “As teach-ers, we must learn to carefully choose the language pathways that support

CONFUSING TERMS

CONFUSING FORMATSanalog and digital clocksangle rotationquadrant layoutsuperscripts and subscriptsvarious types of graphs

CONFUSING SYMBOLS

altitudeanybasecombinationcompute and computercongruent and equivalentdifferencedivide by and divide intodividendequal and equivalentexampleextremefactorfact

imaginarylimitmean and medianmultiplesnumber and numeralof and offoperationor (exclusive) vs. or

(inclusive)originpipowerprimeproduct

radicalrangereflectionregularrelationshipremainder (division) vs.

remainder (subtraction)right angle and left anglesimilarsine and signsum and sometangentvariable

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mathematical understanding, and simultaneously, we must be alert for lan-guage pitfalls that contribute to misunderstandings of mathematical ideas.More specifically, we must learn how to invite, support, and model thought-ful explanation, evaluation, and revision of mathematical ideas using correctmathematical terms and symbols” (p. 56).

The intent of this book is to facilitate this invitation, this support, and thismodeling by opening classroom doors and sharing the wisdom of teacherswho have reflected deeply on how best to create and extend the mathemati-cal fluency of their students.

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2

9

The students know how to do the math, they just don’t understand what the question is asking.

The thing I don’t like about this new series is the way the problems are stated;they’re hard for the students to get what to do.

The reading level is too hard for the students.

I have to simplify, to reword the questions for my students, and then they can do it.

In my three years working as a mathematics coach to 6th, 7th, and 8th gradeteachers, I’ve often heard statements such as these. There seems to be an ideathat somehow it is unfair to expect students to interpret problems on stan-dardized tests and in curriculum texts: after all, what does evaluating studentreading skills have to do with mathematics? When people I meet find out thatI teach mathematics, they often say, “I did OK in math—except for thoseword problems.”

To many teachers, mathematics is simply a matter of cueing up proceduresfor students, who then perform the appropriate calculations. Over and over,I hear teachers interpret problems for their students when asked what a ques-tion means or when a student says, “I don’t know what to do.” This startedme thinking about the mathematics teacher’s role in helping students tointerpret problems.

Certainly teachers try to help students to read and interpret mathematics textand discuss problem-solving strategies with them. I hear them say such thingsas “of means times” and “total means you probably have to add something.”However, when you think about it, most strategies are still procedural—“follow

Reading in the Mathematics ClassroomDiana Metsisto

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this recipe”—rather than about helping students to read for understanding (i.e.,to interpret text and to reason).

Unless mathematics teachers are generalists and have been trained in read-ing instruction, they don’t see literacy as part of their skill set. More impor-tant, they don’t appreciate that reading a mathematics text or problem isreally very different from other types of reading, requiring specific strategiesunique to mathematics. In addition, most reading teachers do not teach theskills necessary to successfully read in mathematics class.

Listening to teachers reword or interpret mathematics problems for theirstudents has led me to start conversations with teachers about taking time towork specifically on reading and interpretation. One strategy we arrived at isfor teachers to model their thinking out loud as they read and figure out whata problem is asking them to do. Other strategies include dialoguing with stu-dents about any difficulties they may have in understanding a problem andasking different students to share their understanding. The strategies that wehave shared have come from years of working in the classroom to improvestudent comprehension. None of us had previously studied the unique diffi-culties involved in reading mathematics texts.

All mathematics teachers recognize the need to teach their students toread and interpret what I’ll call mathematical sentences: equations andinequalities. The National Council of Teachers of Mathematics (1996) statesthat, “[b]ecause mathematics is so often conveyed in symbols, oral and writ-ten communication about mathematical ideas is not always recognized as animportant part of mathematics education. Students do not necessarily talkabout mathematics naturally; teachers need to help them to do so” (p. 60).Knowing how to use the unique symbols that make up the shorthand ofmathematical statements—such as numerals, operation signs, and variablesthat stand in for numbers—has always been part of what mathematics teach-ers are expected to teach. So in a limited way, we have always been readingteachers without realizing it.

Martinez and Martinez (2001) highlight the importance of reading to math-ematics students:

[Students] . . . learn to use language to focus and work through problems,to communicate ideas coherently and clearly, to organize ideas and struc-ture arguments, to extend their thinking and knowledge to encompassother perspectives and experiences, to understand their own problem-solving and thinking processes as well as those of others, and to develop

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11Reading in the Mathematics Classroom

flexibility in representing and interpreting ideas. At the same time, theybegin to see mathematics, not as an isolated school subject, but as a lifesubject—an integral part of the greater world, with connections to con-cepts and knowledge encountered across the curriculum. (p. 47)

James Bullock (1994) defines mathematics as a form of language inventedby humans to discuss abstract concepts of numbers and space. He states thatthe power of the language is that it enables scientists to construct metaphors,which scientists call “models.” Mathematical models enable us to think criti-cally about physical phenomena and explore in depth their underlying ideas.Our traditional form of mathematics education is really training, not educa-tion, and has deprived our students of becoming truly literate. Knowing whatprocedures to perform on cue, as a trained animal performs tricks, is not thebasic purpose of learning mathematics. Unless we can apply mathematics toreal life, we have not learned the discipline.

If we intend for students to understand mathematical concepts rather thanto produce specific performances, we must teach them to engage meaning-fully with mathematics texts. When we talk about students learning to readsuch texts, we refer to a transaction in which the reader is able to ponder theideas that the text presents. The meaning that readers draw will dependlargely on their prior knowledge of the information and on the kinds ofthinking they do after they read the text (Draper, 2002): Can they synthesizethe information? Can they decide what information is important? Can theydraw inferences from what they’ve read?

Reading Requirements for Mathematics Text

Let’s look at some ways in which mathematics text differs from text in othersubjects. Research has shown that mathematics texts contain more conceptsper sentence and paragraph than any other type of text. They are written ina very compact style; each sentence contains a lot of information, with littleredundancy. The text can contain words as well as numeric and non-numericsymbols to decode. In addition, a page may be laid out in such a way that the eye must travel in a different pattern than the traditional left-to-right oneof most reading. There may also be graphics that must be understood for thetext to make sense; these may sometimes include information that is intendedto add to the comprehension of a problem but instead may be distracting.

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Finally, many texts are written above the grade level for which they areintended (Barton & Heidema, 2002).

Most mathematics textbooks include a variety of sidebars containing proseand pictures both related and unrelated to the main topic being covered. Inthese we might find a mixed review of previous work, extra skills practice, alittle vignette from an almanac, a historical fact, or a connection to some-thing from another culture. Such sidebars often contain a series of questionsthat are not part of the actual exercises. Although they are probably added togive color and interest to the look of the page, they can be very confusing toreaders, who might wonder what they are supposed to be paying attention to.Spending time early in the year analyzing the structure of the mathematicstextbook with students can help them to read and comprehend that text.

When I reflect on my own experiences in the classroom, I realize that stu-dents need help finding their way around a new text. They often will just readone sentence after another, not differentiating among problem statements,explanatory information, and supportive prose. As we strive to develop inde-pendent learners, asking students questions about the text structure can helpthem to focus on the idea that texts have an underlying organization, thatdifferent texts may have different structures, and that it is important to ana-lyze the structure of the text being used.

In addition to the unique page formatting and structure of most mathe-matics texts, the basic structure of mathematics problems differs from that ofmost informational writing. In a traditional reading paragraph, there is atopic sentence at the beginning and the remaining sentences fill in detailsthat expand on and support this main idea; in a mathematics problem, thekey idea often comes at the end of the paragraph, in the form of a questionor statement to find something (e.g., “How many apples are left?” “Find thearea and perimeter of the figure above.”). Students must learn to read throughthe problem to ascertain the main idea and then read it again to figure outwhich details and numbers relate to the question being posed and which areredundant. Students have to visualize the problem’s context and then applystrategies that they think will lead to a solution, using the appropriate datafrom the problem statement.

Some of the symbols, words, notations, and formats in which numbersappear can be confusing. For instance, when do you use the word number asopposed to numeral? Do you indicate numbers with numeric symbols, or withwords? The term remainder can be used in problems solved by both division

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and subtraction. The equal sign can represent quantities of exactly the samevalue, or items that are equivalent.

I have seen students read 12:10 on a digital clock and interpret it as mean-ing 121⁄10 hours instead of 121⁄6. This illustrates the difficulty of using digitalclocks to help students picture elapsed time: digital clocks only present uswith digits for isolated moments in time, whereas analog clocks—with theircircular faces, hand angles, and pie wedges—provide a concrete model of thefractional parts of an hour, thus adding to our understanding of how time isdivided.

Same Words, Different Languages

Adding to the confusion of this dense language of symbols is the fact thatmany mathematical terms have different meanings in everyday use. Forexample, the word similar means “alike” in everyday usage, whereas in math-ematics it means that the ratios of the corresponding sides of two shapes areequivalent and corresponding angles are equal. Thus in everyday English, allrectangles are “similar” because they are alike, whereas in mathematics theyare “similar” only if the ratio of the short sides equals the ratio of the longsides. Mathematical terms such as prime, median, mean, mode, product, combine,dividend, height, difference, example, and operation all have different meaningsin common parlance.

In addition to words, mathematical statements and questions are alsounderstood differently when made in a non-mathematical context. For exam-ple, right angles are often drawn with one vertical line and with one perpen-dicular line extending from it to the right. When shown a right angle with theperpendicular line extending to the left, a student once asked a colleague ofmine, “Is that a left angle?”

According to Reuben Hersh (1997), students must be taught that the lan-guage we read and speak in mathematics class is actually a technical jargon,even though it may look and sound like regular English. For example, zero isnot really a number in everyday language—when we say we have “a numberof books” in English, we never mean zero (or one, for that matter). But inmathematics, 0 and 1 are both acceptable answers denoting the concept of “anumber.” Similarly, when we “add” something in English, we invariablymean that we are increasing something. In mathematics, however, addition

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can result in an increase, a decrease, or no change at all depending on whatnumber is being added. Hersh adds the following example: The answer to the question, “If you subtract zero from zero, what’s the difference?” is, inmathematics, zero. We are explicitly asking for a numerical answer. But inEnglish, the question can be interpreted as, “Who cares?” (i.e., “What’s thedifference?”).

When a girl in a class I was observing was asked, in reference to a city map,“How might you go from City Hall to the police station?” she replied, “By car, walk—I don’t know.” She understood “how” to mean “by what means”rather than “by following what path.” This student is not alone in findingsuch mathematics questions puzzling.

Small Words, Big Differences

In English there are many small words, such as pronouns, prepositions, andconjunctions, that make a big difference in student understanding of mathe-matics problems. For example:

• The words of and off cause a lot of confusion in solving percentageproblems, as the percent of something is quite distinct from the percent offsomething.

• The word a can mean “any” in mathematics. When asking students to“show that a number divisible by 6 is even,” we aren’t asking for a specificexample, but for the students to show that all numbers divisible by 6 have tobe even.

• When we take the area “of” a triangle, we mean what the students thinkof as “inside” the triangle.

• The square (second power) “of” the hypotenuse gives the same numericalvalue as the area of the square that can be constructed “on” the hypotenuse.

A study by Kathryn Sullivan (1982) showed that even a brief, three-week pro-gram centered on helping students distinguish the mathematical usage of“small” words can significantly improve student mathematics computationscores. Words studied in the program cited by Sullivan include the, is, a, are, can,on, who, find, one, ones, ten, tens, and, or, number, numeral, how, many, how many,what, write, it, each, which, do, all, same, exercises, here, there, has, and have.

I remember once observing a lesson on multiplying fractions using an areamodel. The teacher had asked me to script her “launch”—the segment of thelesson designed to prepare students for a paired or small-group exploration of

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the topic. Because the teacher felt that the mathematics textbook was too dif-ficult for her students, she read the text aloud and asked students to restatewhat she said in their own words. My notes show that the teacher spoke in a soft, conversational tone. She clearly enunciated the content vocabularyrequired for the lesson and clarified the meanings of nouns and adjectivesrelated to the topic, and of the verbs for the procedures necessary to completethe activity. However, my notes also showed that some pronouns hadambiguous referents (e.g., “You multiply it . . .”) and that the teacher’s softtone made some prepositions barely audible. For example, the text asked stu-dents to find half of 21⁄4 pans of brownies (the teacher read it as “two and afourth”). If they weren’t following along in their books, what did the studentshear—“two ‘nda fourth” or “two ‘nta fourths”? As a matter of fact, one stu-dent took a sheet of notebook paper and wrote “2/4” at the top. Next, he drewtwo squares. Finally he used horizontal lines to divide each square intofourths. I pointed to the “2/4” and asked what it meant. He replied, “Two-fourths is two pans divided into fourths.” And to that particular student, halfof that quantity was one.

Enunciating small but significant words more precisely, being more awareof the confusion that these words can engender, and emphasizing the correctuse of these little land mines will not only enhance computational skills, butalso help students answer open-response questions more accurately.

Strategic Reading

Literacy researchers have developed some basic strategies for reading to learn.Here is a summary of strategies outlined by Draper (2002):

Before reading, the strategic reader• Previews the text by looking at the title, the pictures, and the print inorder to evoke relevant thoughts and memories• Builds background by activating appropriate prior knowledge aboutwhat he or she already knows about the topic (or story), the vocabulary,and the form in which the topic (or story) is presented• Sets purposes for reading by asking questions about what he or shewants to learn (know) during the reading episode

While reading, the strategic reader• Checks understanding of the text by paraphrasing the author’s words

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• Monitors comprehension by using context clues to figure out unknownwords and by imagining, inferencing, and predicting• Integrates new concepts with existing knowledge, continually revisingpurposes for reading

After reading, the strategic reader• Summarizes what has been read by retelling the plot of the story or themain idea of the text• Evaluates the ideas contained in the text• Makes applications of the ideas in the text to unique situations, extend-ing the ideas to broader perspectives. (p. 524)

Mathematics teachers can use this general outline in several ways. They canmodel the process by reading the problem out loud and paraphrasing theauthor’s words and then talking through how they use context clues to figureout word meanings. Before reading, teachers can ask questions that they wantstudents to consider as they approach a mathematics problem. Teachers canprobe about the reading’s vocabulary by asking questions such as, “Are weclear on the meaning of all of the words?” or “Does the context help or shouldwe look the word up?” Also significant are questions about the meaning of theproblem, such as, “Can I paraphrase the problem?” “Does the problem makesense to me?” or “Does my understanding incorporate everything I’ve read?”

Reinforcing the idea that a piece of mathematics text needs to make sense(and that it can make sense) is exceedingly important. Teachers need to pro-vide explicit scaffolding experiences to help students connect the text to theirprior knowledge and to build such knowledge. In her book Yellow Brick Roads(2003), Janet Allen suggests that teachers need to ask themselves the follow-ing critical questions about a text:

• What is the major concept?• How can I help students connect this concept to their lives?• Are there key concepts or specialized vocabulary that needs to be intro-

duced because students could not get meaning from the context?• How could we use the pictures, charts, and graphs to predict or antici-

pate content?• What supplemental materials do I need to provide to support reading?

Consider the following three situations I encountered while working withtwo 6th grade mathematics teachers and an 8th grade mathematics teacher:

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• In the first case, the 6th grade teacher was explicitly teaching studentshow to look for context clues. The question at hand required students to write“7 � 7 � 7 � 8 � 9 � 9” in exponential notation. The teacher suggested thatthe students look for a word in the text of the question that might help them.It was interesting to see the different ways that students interpreted thissimple exercise. Some seemingly did not look at the words at all; they simplyexecuted the calculation. Some knew the word notation and knew that writemeant to reformat the problem. Those who also knew the word exponent wereable to answer the problem correctly, whereas some of those who didn’t knowwhat exponent meant used a different type of notation to rewrite the problemin words. It is clear that simple exercises such as these can help students tointerpret mathematics text by looking at all the words, rather than assumingthat a calculation is always sought.

• In the second case, students in a 6th grade class were asked to find thepercentage of cat owners who said their cats had bad breath. In a survey, 80out of 200 cat owners had said yes. The students used several different strate-gies to answer the question and discussed it as a class. They were then askedto read and answer some follow-up questions. The first one read, “If you sur-vey 500 cat owners, about how many would you expect to say that their catshave bad breath? Explain your reasoning.” The students asked the teacher tohelp them understand what was being asked, and she complied, as teachersoften do without thinking, by telling the students to use the 40 percent fig-ure from the previous question.

If we are really trying to help students read and understand for themselves,we must ask them questions instead of explicitly telling them what the textmeans: “What information do you have that might help you answer this ques-tion?” “Does the fact that this is a ‘follow-up’ help us to decipher the question?”

• In the third case, groups of 8th graders worked through a series of prob-lems involving compound interest calculations. The main question read,“What are the initial value, rate of increase in value, and number of years thatSam is assuming?” In one group, a boy said loudly, “It’s too hard to figure out!I don’t want to use my brain,” and snapped his book shut. When questionedabout his reaction, the boy just said, “Too many questions.”

Students often have difficulty with this sort of multipart question. Theyneed to develop the simple strategy of taking the main question apart andlisting the individual questions separately.

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As a teacher, I often had students come to me for help understanding aproblem. Just asking them to read the problem aloud would usually elicit,“Oh, now I get it.” My experience suggests that having students read prob-lems aloud to themselves can help their understanding. I also think that, forsome students, the attention of someone else listening may help them tofocus.

Other Reading Strategies

In addition to helping students learn the meaning in mathematics text of “lit-tle” mathematics words and the precise mathematical meanings of familiarEnglish words, teachers should help them understand the abstract, unfamil-iar terminology of mathematics. As stated by Barton, Heidema, and Jordan(2002), and as I’ve learned from my own experience in the classroom, just giv-ing students vocabulary lists with definitions, or asking them to look up thedefinitions, isn’t enough for them to develop the conceptual meaning behindthe words or to read and use the vocabulary accurately.

Teachers can also introduce various maps, webs, and other graphic orga-nizers to help students further organize mathematics meanings and concepts.Two graphic organizers that can be particularly useful in mathematics classesare the Frayer Model (Frayer, 1969) and the Semantic Feature Analysis Grid(Baldwin, 1981). In the Frayer Model, a sheet of paper is divided into fourquadrants. In the first quadrant, the students define a given term in their ownwords; in the second quadrant, they list any facts that they know about theword; in the third quadrant, they list examples of the given term; and in the fourth quadrant, they list nonexamples. (See Figure 2.1 for an example of the Frayer Model.)

The Semantic Feature Analysis Grid helps students compare features ofmathematical objects that are in the same category by providing a visualprompt of their similarities and differences. On the left side of the grid is a listof terms in the chosen category, and across the top is a list of properties thatthe objects might share. (An example of the Semantic Feature Analysis Grid isshown in Figure 2.2.)

Another useful problem-solving process is the SQRQCQ process, developedby Leslie Fay (1965), which is a variation of Polya’s four-step process (1973).The acronym SQRQCQ stands for the following terms and respective actions:

• Survey. Read the problem quickly to get a general understanding of it.• Question. Ask what information the problem requires.

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• Read. Reread the problem to identify relevant information, facts, anddetails needed to solve it.

• Question. Ask what operations must be performed, and in what order, tosolve the problem.

• Compute/Construct. Do the computations, or construct the solution.• Question. Ask whether the solution process seems correct and the answer

reasonable.

FIGURE 2.1 Sample Frayer Model for Composite Numbers

Definition

A whole number with more than twofactors.

Facts

• 4 is the lowest composite.• 0 and 1 are not composites.• Square numbers have an odd number

of factors.• 2 is the only even number that is not a

composite.

Examples

4, 6, 8, 9, 10, 12, 14, 15, 16

Nonexamples

0, 1, 2, 3, 5, 7, 11, 13, 17

FIGURE 2.2 Sample Semantic Feature Analysis Grid for Quadrilaterals

Term Sides Angles Opposite Opposite Only One FourEqual Equal Sides Sides Pair of Sides

Parallel Equal ParallelSides

Parallelogram X X X

Rectangle X X X X

Rhombus X X X X

Scalene Quadrilateral X

Square X X X X X

Trapezoid X X

CompositeNumbers

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Teachers can model the steps for the students with a chosen problem andthen have the students practice individually or in pairs. Students can then beasked to share their use of the strategy with a partner, within a group, or withthe class.

Elementary Classroom Issues

Most elementary teachers teach mathematics as one of several subjects; inmany cases, they teach reading as well as mathematics, unlike teachers inmiddle school and high school. They need to be aware of the particular diffi-culties involved in reading mathematical text. When encountering mathe-matical symbols, students face a multilevel decoding process: First they mustrecognize and separate out the confusing mathematical symbols (e.g., +, �, <)without any phonic cues; then they must translate each symbol into English;and finally they must connect the symbol to the concept for which it standsand then carry out the operations indicated.

Graphs and Tables

Graphs are also particularly hard for elementary students to read. The firsttype of graph that most students encounter is the bar graph, which is mostcommonly “read” from bottom to top. There are many types of graphs—particularly in the mathematics, science, and social studies contexts—withdifferent “directions of readability.”

I became aware of the need to help students learn to stop and analyzegraph and table structures when working with (what I thought were) simplematrix puzzles, involving only two rows and two columns, with an opera-tion sign in the upper left corner. The numbers at the top and to the left were to be combined using the operation sign, and the answers were to bewritten in the interstices of the rows and columns. The idea was for the stu-dent to fill in any missing cells in the matrix. (Figure 2.3 is an example of acompleted puzzle.)

Several students had difficulty understanding what they were expected todo with the puzzle: What was to be added? Where did the answer go? Thisexperience pointed out to me that specific strategies to decipher graphic rep-resentations need to be extensively modeled and repeatedly explored. It isimportant that students become aware that an underlying plan or pattern canusually be discovered by careful study.

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Guided Reading One strategy that may be familiar to elementary reading teachers, and whichseems particularly useful in the context of mathematics, is that of guidedreading sessions (Allen, 2003). In such sessions, the teacher is still responsiblefor helping students connect what they are reading to prior knowledge. Theteacher should first present the text or graphic to students in small, coherentsegments, being sure to process each segment before going on to the nextone. As the reading progresses, the teacher should ask process questions thatshe wants the students to ask themselves in the future. They may be asked topredict what the reading will be about simply by reading the title of the piece(if there is one, such as a graph or story problem). Next the students shouldmake two columns on a piece of paper, one headed “What I Predict” and theother headed “What I Know.” Once the students have silently read each sec-tion of the piece, they should fill out each column accordingly. At this point,the teacher should ask students questions such as the following:

• What would you be doing in that situation?• Does this make sense?• What does the picture/graph/chart tell you?• How does the title connect to what we’re reading?• Why are these words in capital letters?• Why is there extra white space here?• What does that word mean in this context?

Figure 2.4 shows a simple example of a possible guided reading for a lessonfrom an algebra text. The text would be unveiled one paragraph (or equation)at a time rather than given to the students as one continuous passage.

FIGURE 2.3 Sample Matrix Puzzle

+ 2 6

3 5 9

9 11 15

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FIGURE 2.4 Guided Reading Example

TEXT POSSIBLE QUESTIONS

Solving Systems Using Substitution 1. What does the title tell you?

Problem

From a car wash, a service club made $109 2. Before you read further, how would you that was divided between the Girl Scouts and translate this story problem into equations?the Boy Scouts. There were twice as many girls as boys, so the decision was made to give the girls twice as much money. How much did each group receive?

Solution

Translate each condition into an equation. 3. What do they mean here by “condition”? Suppose the Boy Scouts receive B dollars and the Girl Scouts receive G dollars. We number the equations in the system for reference.

The sum of the amounts is $109. 4. Did you come up with two equations (1) B + G = 109 in answer to question 2 above? Are the

Girls get twice as much as boys. equations here the same as yours? If not, (2) G = 2B how are they different? Can you see a

way to substitute?

Since G = 2B in equation (2), you can substitute 2B for G in equation (1).

B + 2B = 109 5. How did they arrive at this equation?3B = 109 6. Do you see how it follows?

B = 36 1/3 7. Does it make sense? How did they get this?

To find G, substitute 36 1/3 for B in either 8. Do this, then we’ll read the next part.equation. We use equation (2).

G = 2B= 2 � 36 1/3= 72 2/3

So the solution is (B, G) = (36 1/3, 72 2/3). 9. Did you get the same result?The Boy Scouts will receive $36.33, and the Girl Scouts will get $72.67.

Check

Are both conditions satisfied? 10. What conditions do they mean here?

Will the groups receive a total of $109? 11. How would you show this?Yes, $36.33 + $72.67 = $109. Will the boys Where did they get this equation?get twice as much as the girls? Yes, it is as close as possible.

Note: Text in the left column above is adapted from University of Chicago School Mathematics Project: Algebra (p. 536), by J. McConnell et al., 1990, Glenview, IL: Scott Foresman.

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Guided reading is best done in small groups, with the teacher encouragingstudents to think of their own questions as they read. A predetermined set ofquestions isn’t necessary. The purpose of guided reading is to help studentsrealize that they can engage with and make sense of the text, whether it be inlanguage arts or mathematics.

Conclusion

Mathematics teachers don’t need to become reading specialists in order tohelp students read mathematics texts, but they do need to recognize that stu-dents need their help reading in mathematical contexts. Teachers shouldmake the strategic processes necessary for understanding mathematics explicitto students. Teachers must help students use strategies for acquiring vocabu-lary and reading word problems for meaning. Students are helped not by hav-ing their reading and interpreting done for them, but rather by being askedquestions when they don’t understand the text. The goal is for students tointernalize these questions and use them on their own.

Mathematics teachers are ultimately striving to help their students under-stand mathematics and to use it in all aspects of their lives. Being aware thatstudents’ prior knowledge and background affects their comprehension isvastly important, as is explicitly analyzing the organization of mathematicstexts. When we share strategies for understanding text, question our stu-dents about their conceptual processes, and model strategies and question-ing techniques, we are helping students to develop metacognitive processesfor approaching mathematics tasks. Mathematics teachers should recognizethat part of their job in helping their students become autonomous, self-directed learners is first to help them become strategic, facile readers of mathe-matics text.

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24

Dave has taught mathematics for twelve years. Before that, he had been anaccountant, but he felt he wasn’t doing enough with his life, so he went backto college to be certified as a teacher. When hired to teach 7th grade, he threwhimself into his work, applying all his organizational talents to the classroom.He kept track of the skills and concepts that he needed to teach and moni-tored each student’s progress. Over time, he became a resource to the othermathematics teachers because of his content knowledge (he was certified toteach through 12th grade), his organization, and his understanding of math-ematics in the real world.

Despite his success, Dave was disappointed in his students’ struggles witha unit on fractions, decimals, and percentages. He decided to put extra timeinto the unit during the first half of the year, and to review the material inwarm-ups—five- to ten-minute practice sessions at the beginning of eachclass—the rest of the year. He insisted that the students who struggled themost work with him after school. Yet when the state assessment scores cameout, Dave was dismayed—many of his students did not do well on questionsinvolving fractions, decimals, and percentages. He discussed the results withhis principal and colleagues and attributed his students’ low scores to theirpoor attitudes toward mathematics, their low work ethic, and their unwilling-ness to do homework.

Writing in the Mathematics ClassroomCynthia L. Tuttle

Note: This chapter contains material that has been released to the public by the MassachusettsDepartment of Education. The Department of Education has not endorsed the material containedin this chapter.

3

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25Writing in the Mathematics Classroom

The next year, Dave had the opportunity to become an 8th grade teacher.He jumped at the chance to teach “harder” mathematics; he also thought he would have a head start with the students, as he had taught most of themthe previous year. Dave opened the year with a pretest to identify students’strengths and weaknesses. When he corrected the pretests, he found that thestudents with whom he had worked so diligently the year before had forgot-ten much of what they’d “learned” about fractions, decimals, and percent-ages. He was again baffled and discouraged. Despite his conscientious teaching,the students were not retaining the material or successfully applying it to newsituations.

It’s not unusual for mathematics teachers to become frustrated becausetheir students were not well-enough prepared in earlier grades for what theymust now learn. Talk to the prior mathematics teachers, and they will main-tain that they did what they could. So the question becomes: If students havebeen taught the material and haven’t learned or retained it, what can we asprofessionals do to change the scenario?

In searching for answers to this question, I began to investigate the role ofwriting in the mathematics classroom. For me, writing has always been impor-tant to my learning; I write out everything from professional workshop pre-sentations to lists of what I want to discuss with the doctor. Writing in thisway slows down and focuses my thinking; I am able to hear each word in myhead and see it on paper. It is like a mindful meditation during which I shut outthe rest of the world and am totally engaged in the process.

Another benefit of writing is that it allows the page to become a holdingplace for our thoughts until we can build upon them. We can revisit our writ-ten thoughts as often as needed and thus revise our thinking. Although I start with an overall plan when I write, I do not know where the ideas andwords will take me until the process of writing drags them out of me—muchas many artists do not know where a picture is going until the paint touchesthe canvas.

Linguistic Versus Logical-Mathematical Intelligence

Throughout life, we try to make sense of the world using the four skills of lis-tening, speaking, reading, and writing. According to Howard Gardner (1983),these skills make up our linguistic intelligence and form the basis for much ofour school achievement. There is, however, another intelligence that alsoaffects our achievement in school: logical-mathematical intelligence. The

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main difference between these two intelligences, notes Gardner, is that lin-guistic intelligence is not closely tied to the realm of physical objects, whereaslogical-mathematical intelligence is primarily rooted in the material world.Both types of intelligences have been the focus of most “achievement” test-ing in the United States.

Traditionally, linguistic intelligence has been considered more importantthan logical-mathematical intelligence. Students with poor linguistic intelli-gence are often considered less capable than their peers and are placed inlower-level groups or classes. If this practice continues through secondaryschool, it begins to limit the type of higher education available to the strug-gling students. As adults, many of them will be forced to devise involvedstrategies to hide their literacy problems.

Failure in mathematics does not carry as severe a stigma. In some circles, itis viewed in the same way as not being able to carry a tune—lamentable, butnot overly significant. However, as our economy becomes more and moretechnology-based, an imperfect sense of mathematical pitch can have seriousramifications. As with their literacy-challenged peers, students who do not dowell in mathematics are often placed in lower groups or classes, thus restrict-ing their future academic choices. Fortunately, some of this is starting tochange. Mathematics is beginning to be viewed less as a series of arithmeticcalculations than as “the science of order, patterns, structure, and logical rela-tionships” (Devlin, 2000).

Changing Perceptions and New Expectations

In its Principles and Standards for School Mathematics (2000), the National Coun-cil of Teachers of Mathematics (NCTM) directs that mathematics programs bedesigned so that all students can succeed. Whereas mathematics used to beviewed as a narrow topic with limited student success rates, it is now conceivedas broad and requiring the understanding of all students. This is a momentouschange of perception, and an equally momentous challenge for educators.

As Zinsser stresses in his book Writing to Learn (1989), it is important thatall students be involved in the mathematics classroom. Twenty-five studentscannot all speak at the same time, but they can all write at the same time, andwriting encourages them to become engaged in their learning.

I will never forget a parent-teacher conference early in my career. The stu-dent under discussion had scored poorly in mathematics; though he could

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hold all the big ideas in his head, he struggled mightily when it came to com-putation. For some reason, he just couldn’t get the numbers on paper cor-rectly. I knew I had to share these difficulties with his parents, who Isuspected had heard this chronicle of failure before. I desperately wanted himto succeed—of all my students, he displayed the greatest discrepancy betweenachievement in computation and understanding of mathematics concepts. AsI started to describe his progress so far, his mother interrupted me and said,“Oh, don’t worry about his mathematics. I was the same way in school, andnow I manage a $7 million budget just fine.” I have thought a lot about hercomment since then, and my reflection invariably leads to the question ofwhy can’t we make students like this one as successful in “school math” asthey may become in later life.

Fortunately, mathematics content and pedagogy are changing. The expandedcontent that emphasizes solving real-world problems would have held greatappeal for my calculation-poor, concept-rich student. If his mathematicswork had involved more writing, he would have been better able to empha-size his conceptual strength and resolve his computational difficulties (withthe aid of a two-dollar, four-function calculator if necessary!). Written expla-nations in mathematics are about what is being done and why it works. Thetype of thinking involved in justifying a strategy or explaining an answer isquite different from that needed to merely solve an equation. The process ofwriting about a mathematics problem will itself often lead to a solution.

Written Responses to Mathematics Problems

Early in their schooling, many students are introduced to response logs inlanguage arts class (Maloch, 2002). In their logs, students write commentsand questions about what they have read before engaging in small-groupdiscussions. This practice gives all students the opportunity and responsibilityto interact with what they’ve read. The same experience can be provided in mathematics class. Once students have done some initial writing about aproblem, they can share their strategies in small groups. In attempting tosolve the problem, the students will have additional opportunities for writ-ing. Following each group’s report, the teacher may present a related writingassignment.

If students begin the problem on their own, they are starting from theirown mathematical way of thinking. Bringing their written solutions to the

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FIGURE 3.1 Four Possible Written Responses to a Mathematics Problem

The ProblemFour students could not go on a field trip. They had paid $32 in total for their tickets. How muchmoney was returned to each student?

Directions to the Student

1. Read the problem.2. Write down one possible strategy to solve the problem. Use diagrams or pictures when

possible. 3. Write down any questions you have about the problem.

Possible Response #1

Student Student Student Student#1 #2 #3 #4

$5 $5 $5 $5$2 $2 $2 $2$1 $1 $1 $1

$8 $8 $8 $8

I knew that each student could not get $10 because 4 x $10 = $40, so I returned $5 to each studentfor a total of $20 (4 x $5 = $20). I had $12 left ($32 - $20 = $12). I gave each student $2 (4 x $2 =$8). I had $4 left ($12 - $8 = $4). I gave each student $1 (4 x $1 = $4), and I had returned all themoney ($4 - $4 = $0). Each student received $8 back.


I knew that $32 had to be shared equally among four students.$32 – $4 = $28 Each student received $1.$28 – $4 = $24 Each student received $1.$24 – $4 = $20 Each student received $1.$20 – $4 = $16 Each student received $1.$16 – $4 = $12 Each student received $1.$12 – $4 = $8 Each student received $1.$8 – $4 = $4 Each student received $1.$4 – $4 = $0 Each student received $1.

All the money has been returned. Each student has received a total of $8.


small group helps students investigate mathematics more deeply. There aremany ways that students may respond to this problem. Figure 3.1 shows anexample of a basic problem that a teacher might give to students to assesstheir understanding of the factors of 32, along with four different ways thatstudents might approach it.

(Figure continues on next page)

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By the time the students have finished studying a problem like the one inFigure 3.1, they have looked at it in depth, first from their own perspective andthen from the perspectives of others. Students need to untangle what is intheir own minds first, get it on paper, and then share their thinking with oth-ers. This ensures that there will be a wide range of responses to each question.

FIGURE 3.1Four Possible Written Responses to a Mathematics Problem (Continued)


I knew that $32 had to be shared equally among four students, so I gave each student $1 until all mybills were gone. I gave out $4 at a time.

Time 1: ($1 + $1 + $1 + $1)

Time 2: ($1 + $1 + $1 + $1)

Time 3: ($1 + $1 + $1 + $1)

Time 4: ($1 + $1 + $1 + $1)

Time 5: ($1 + $1 + $1 + $1)

Time 6: ($1 + $1 + $1 + $1)

Time 7: ($1 + $1 + $1 + $1)

Time 8: ($1 + $1 + $1 + $1)

I gave out $4 ($1 + $1 + $1 + $1) at a time.$4 + $4 + $4 + $4 + $4 + $4 + $4 + $4.I gave out $4 eight times. Each student received $8.


I gave out $1 at a time to each of the four students. All together, I have eight groups of $4. Thatmeans each student received $1 eight times for a total of $8.

$

$

$

$

$

$

$

$

$ $ $ $

$ $ $ $

$ $ $ $

$ $ $ $

$ $ $ $

$ $ $ $

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Students absorb mathematics from their peers when they share these responseswith each other. After viewing other student responses, the students can beencouraged to generate additional possible solutions. To quote Stigler andHiebert (1999):

When this type of learning experience is used, the range of individual dif-

ferences will be revealed. Individual differences are beneficial for the class

because they produce a wide range of ideas and solution methods that pro-

vide the material for students’ discussion and reflection. The variety of

alternative methods allows students to compare them and construct con-

nections among them. It is believed that all students benefit from the vari-

ety of ideas generated by their peers. (p. 94)

Mathematics writing requires hard work and planning on the part of theteacher and takes time and practice to develop properly. Teachers who facili-tate this type of learning know their students well and can anticipate possibleresponses based on their students’ prior ones. New variations will likely emergeas students reflect on the solutions of their classmates.

In order for mathematics writing to be effective, the following guidelinesmust be observed:

• The problem must be appropriate for the students who are going to bewriting about it.

• The students must know how to use blocks, diagrams, pictures, or gridsto work out their solutions before writing about them.

• The students must have confidence in their ability to respond to theproblem as individuals. They must think of themselves as successful mathe-matics learners.

• The students must feel comfortable sharing their answers without fear ofbeing ridiculed. This means that the teacher and other students have toaccept all responses as worthy of discussion.

• The problem must be discussed with the whole class, and all strategiesmust be reported.

Additional Writing Strategies

All writing provides valuable opportunities for rewriting as a learning strategy.Students can develop their written responses more fully using cues from class-mates or the teacher. This process gives students more time, and thus a deeper

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connection, with the subject. Other writing-to-learn strategies include journalkeeping, creating problems similar to the one being solved, and directedexpository writing. Fisher (2002) notes that resources are now available to helpteachers guide the process of student writing about mathematical thinking.Once such writing is completed, students’ responses should be arranged in athree-ring binder by category: class notes, developing vocabulary, homework,formal and informal assessments, and journal writing. The binder provides awritten record of students’ thinking and real learning.

Remember that this chapter began with a true story that supports the well-known observation that students don’t remember what they learn in mathe-matics. Writing down what they’ve learned forces students to learn thematerial; as the adage goes, you don’t really understand something until youcan explain it to others. It no longer suffices for students to respond, “I don’tknow how I got the answer, I just know it.” They are expected to scaffold theanswer for the audience—the teacher and the other students—in a way thatexplains the answer. This is a controversial point of view. Parents, students,and even some teachers may balk at having to explain how they derived theiranswers. A correct response is all that is necessary in their minds. In today’smathematics classroom, the expectation is that all students, even those whoinstantly know the answer, should be able to justify their responses.

In other words, teachers should use writing to engage students in mathe-matics thinking at the outset of a lesson and continue asking them to puttheir thinking in writing throughout the lesson to refine their thinking. As wewill see in the next section, this process allows teachers to see the why, notjust the how, of the student’s thinking.

Writing as a Prerequisite for Assessment of Student Learning

The objective of any mathematics lesson is for students to be able to applysome aspect of mathematical thinking. Writing provides unique evidenceteachers can use to assess student mathematics skills. The answer to a prob-lem alone does not tell teachers how students are learning: even if the answeris correct (and assuming that the student did the work independently), thestrategy the student followed is unknown. If the answer is incorrect, theteacher does not know at what point the student’s thinking led to an incor-rect response. As the NCTM (2000) states,

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Many assessment techniques can be used by mathematics teachers, includ-ing open-ended questions, constructed-response tasks, selected responseitems, performance tasks, observations, conversations, journals, and port-folios. These methods can all be appropriate for classroom assessment, butsome may apply more readily to particular goals. For example, quizzesusing simple constructed-response or selected-response items may indicatewhether students can apply procedures. Constructed response or perfor-mance tasks may better illuminate students’ capacity to apply mathematicsin complex or new situations. Observations and conversations in the class-room can provide insights into students’ thinking, and teachers can moni-tor changes in students’ thinking and reasoning over time with reflectivejournals and portfolios. (pp. 23–24)

Writing not only provides a measure of student performance, but also sug-gests to the teacher what type of learning experience to present next. Whereare the students in the learning of the material? What should the next lessonhave as its objective? What can I learn from my students’ partially correctapproaches? What must the students understand before we can progress?These are all questions that can be answered by reviewing student writing.

Teachers can further assess student mathematics knowledge by asking open-ended questions. Responses to questions such as “What is division?” revealwhether or not the students have learned the subject procedurally or concep-tually. These types of exercises are helpful both in the early and later grades. Ifstudents are familiar with mathematics writing at the elementary level, thetransition to secondary school will seem natural. If not, teachers will need tofamiliarize the students with the process. If a standards-based curriculum is inplace, daily open-ended questions should already be built into it.

Educators have slowly come to acknowledge the importance of open-endedquestions in preparing students for high-stakes tests. Also useful are anchorpapers from previous years that provide samples of student writing. Studyingthese papers—examining the mathematics involved, discussing the pedagogy,and reviewing responses—helps teachers practice viewing writing as evidenceof learning.

Figure 3.2 shows an open-response question that appeared on a recent 8thgrade high-stakes assessment test. Think about solving this problem yourself.Would you first draw four black tiles and surround them with white tiles sothat you could count the number of white tiles? Would you make a table?Would you use the table to determine the number of white tiles necessary for

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50 black tiles? Would you use the table to develop the equation? Are you ableto make a scatter plot with the axes labeled?

Now compare your approach with the student papers shown in Figures 3.3–3.6. As the responses in these figures make clear, the students are required toapply their skills, prove their solutions, draw generalizations, and make con-nections using words as well as diagrams—all categories of thinking that areclassified at the higher end of commonly held educational objectives. Thistype of standards-based assessment helps students develop confidence intheir mathematical thinking and clearly provides much more information forthe assessor than would a single answer on a multiple-choice test.

Additional Strategies

In order to tap into their prior knowledge, students need to organize whateverthey are writing about. This is particularly important in developing mathemat-ics vocabulary. The guidelines of the Connected Mathematics Project (Lappan,2002) recommend that students record new words as they initially understand

FIGURE 3.2 Sample Open-Ended Problem from an 8th Grade Test

A worker placed white tiles around black tiles in the pattern shown in the three figures below:

a. Based on this pattern, how many white tiles would be needed for 4 black tiles?

b. Based on this pattern, how many white tiles would be needed for 50 black tiles?

c. Make a scatter plot of the first five figures in this pattern showing the relationship betweenthe number of white tiles and the number of black tiles. Be sure to label the axes.

d. Based on this pattern, explain how you could find the number of white tiles needed for anynumber, n, of black tiles. Show and explain your work.

Source: From Release of Spring 2002 Test Items (question #9), by the Massachusetts Department of Education, 2002,Malden, MA: Author. Reprinted by permission.

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FIGURE 3.3 Sample Response to Tile Problem—#1


their meaning from the context in which they are used. For example, studentsmay have a lesson in which they place one-inch squares within a rectangularfigure:

AUTHOR COMMENT

In this example, the student correctly answers the number of white tiles for 4 black tiles and 50 black tiles, shows a scatter plot of the first five tiles, and provides an expression that can beused to determine the number of white tiles for any number of black tiles.

Source: Student responses in Figures 3.3–3.6 from Release of Spring 2002 Test Items (question #9), by theMassachusetts Department of Education, 2002, Malden, MA: Author. Reprinted by permission.

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The students can then count the number of placed squares and record the areaeither as “eight square inches” or as “eight-inch squares.” Their next assignmentcould be to write their own definition of the word area. At this point, a studentmight write, “Area is the number of one-inch squares that fit inside a rectangle.”With subsequent lessons, students can further develop this initial definition toinclude other measures (standard and metric) and shapes. Different ways ofdetermining the area and eventually computing it evolve through multipleexperiences with the word area and its definition. Vocabulary so developed

AUTHOR COMMENT

In this example, the student answers correctly the number of white tiles necessary to surround 4 black tiles and 50 black tiles and partially describes a pattern that can be used to find thenumber of white tiles necessary to surround any number of black tiles. However, the response doesnot show a scatter plot.


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becomes learned. One system of organizing this vocabulary is to have a sheetof paper for each letter of the alphabet with student-derived definitions. This“dictionary” can be transferred to the next year’s class for continuation.

As the NCTM (2000) notes,

It is important to give students experiences that help them appreciate thepower and precision of mathematical language. Beginning in the middlegrades, students should understand the role of mathematical definitions

AUTHOR COMMENT

In this example, the student does not correctly answer how many white tiles would be necessary tosurround 4 black tiles and 50 black tiles. However, the scatter plot is correct. The student does notprovide the general pattern for finding the number of black tile needed.

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AUTHOR COMMENT

In this example, the student does not show the correct response for the number of white tilesneeded to surround 4 black tiles or 50 black tiles and does not show a scatter plot. However, thediagrams for the number of white tiles necessary to surround 1, 2, 3, 4, and 5 black tiles arecorrect, and the labels for each diagram are correct. There is an indication that the student isbeginning to notice a pattern: “every time you add a black one you add two white ones.”


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FIGURE 3.7 Sample Structured Writing Guide

Paragraph One: Problem Statement

Write answers to these two questions:• What is the problem about?• What am I supposed to find?

Paragraph Two: Work Write-Up

Explain step-by-step and in detail everything you did to arrive at each of your answers. Thinkof this as a recipe for someone to follow or as directions to your house. Complete the follow-ing sentences:

• First I . . . • Then I . . . • Next I . . . • After that I . . • Finally I . . .

Paragraph Three: Answer

Prove that your answer is correct by referring to the math you did. Do not write that youchecked it on the calculator, you did it twice, or your friend told you it looked OK. Completethe following sentences:

• My answer is . . . • My answer makes sense because . . .

Source: From Guide for Writing Response in Mathematics, by L. Jubinville, unpublished manuscript, 2002. Reprintedwith permission.


and should use them in mathematical work. Doing so should become per-vasive in high school. However, it is important to avoid a premature rushto impose formal mathematical language; students need to develop anappreciation of the need for precise definitions and for the communicativepower of conventional mathematical terms by first communicating intheir own words. Allowing students to grapple with their ideas and developtheir own informal means of expressing them can be an effective way tofoster engagement and ownership. (p. 63)

Providing a structured guide for their writing can be particularly helpful forstudents. In one urban district, 6th graders responded positively to the struc-tured writing guide in Figure 3.7, which the teacher revised as needed.

Mathematics Writing and ESL Students

Research at California State University indicates that some aspects of learninga second language are similar to learning a first language, and occur naturally

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with a shift to a new language environment. Beyond this natural learning,however, direct instruction is necessary. The grammatical items that tend tocause the most difficulty for English-as-a-second-language (ESL) students arecount and noncount nouns, articles, prepositions, tense, and vocabulary.When students do not know the word they need in the second language, theymust find ways to write around it. Describing processes and concepts is a hugechallenge for ESL students; strategies such as maintaining a vocabulary log,prewriting, editing, and revising can help (California State University WritingCenter, 2003).

There is some debate as to whether or not students should be verbally fluentbefore they begin to write. According to Nekita Lamour of the Hood Children’sLiteracy Project (2003), students should be able to write in their own languagefirst and have it translated into the new language. (Initially someone else shouldtranslate the writing; as the students’ writing skills develop, they may translatetheir work themselves and have it edited.) Lamour also feels it is important thatstudents continue to develop as speakers in their native language.

All students, regardless of their language or cultural background, must studya core curriculum in mathematics based on the NCTM standards. For studentsto truly develop mathematics literacy, however, lessons must be based on real-life experience, involve problem solving, and occur in an environment inwhich students hear, speak, and write mathematics language. For ESL students,teachers must focus on their skills both in mathematics language and inEnglish. For example, if the students are working with obtuse angles, they needto understand the meaning of the suffix -er in “greater than.” They also need tounderstand the connection between angles in mathematics class and angles asdiscussed in science and social studies classes. Teachers should initially keeplanguage simple, particularly if the concept being discussed is hard to grasp. Itis beneficial if the English and mathematics teachers of ESL students train andwork together regularly. Teacher intervention is more important than repetitionand drills; this is as important for ESL learners as it is for regular-track students.

The Cognitive Academic Language Learning Approach

The Cognitive Academic Language Learning Approach (CALLA) for Mathemat-ics and Science provides support for ESL learners in both content and learningstrategies (Spanos, 1993). Figure 3.8 shows three strategies developed by CALLAto help students solve problems successfully. Though specifically designed withESL learners in mind, these strategies are also applicable to all students.

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Using student writing as evidence for assessment purposes is particularlyimportant with ESL students. Figure 3.9 depicts part of an assessment prob-lem from an 8th grade ESL class. Envision that the teacher writes a responseto each answer in order to encourage students to think more deeply about thequestion and to write down more of their thinking.

FIGURE 3.8 Three CALLA Strategies

#1: Word Problem Procedure1. Choose a partner or partners.2. Choose a problem and write it out.3. Have one student read the problem out loud. Discuss the vocabulary and circle words

you don’t understand. 4. Use a dictionary or partner for help, and write out the definitions of the vocabulary words

you did not understand. 5. Write out what the problem is asking you to find.6. Consider what process you should use to solve the problem. Should you add? Subtract?

Multiply? Divide? 7. Solve the problem. 8. Check your answer.9. Explain your answer to your partner.

10. Write your explanation.11. Explain your answer to the class.12. Write a similar problem on a piece of paper.

#2: Mathematics Learning Strategy ChecklistThere are many ways to solve problems. Check the two or three things that you did most whileyou worked on this problem. There are no right or wrong answers.

� I looked for the important words to solve the problem.

� I read the question carefully.

� I remembered how I solved other problems like this one.

� I did the problem in my head because it was easy.

� I formed a picture in my head or drew a picture.

#3: Math Student Self-EvaluationThese are two important things I learned in math today/this week/this month:

1.

(Figure continues on next page)

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Here are some examples of the students’ answers, each with the accompa-nying teacher response:

Student 1: By looking at the graph, you can tell Anne is going the samespeed because it shows every 6 miles: 6, 12, 18, 24, 30, 36 miles. And there isa straight line. It never is a flat line.

Teacher: What can you say about the distance Anne travels each hour?


FIGURE 3.8 Three CALLA Strategies (Continued)

2.

This was an easy problem for me:

This was a difficult problem for me:

I need more help with:

This is how I feel about math today/this week/this month (circle the words that are true):

successful happy excited

confused interested worried

relaxed bored upset

This is where I got help (circle the words that are true):

my teacher

my friend or classmate

my parents

Source: From ESL Math and Science for High School Students, by G. Spanos, paper presented at the 3rd NationalResearch Symposium on Limited English Proficient Student Issues, 1993, Washington, DC: National Clearinghousefor English Language Acquisition.

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FIGURE 3.9 Sample Assessment Problem from an ESL Class

Bicycle Rides

The Liberty Bicycle Path is 36 miles long. The following graph describes Anne’s ride along thebike path one afternoon between 3 p.m. and 6 p.m. How can you tell from the graph thatAnne was riding at the same speed for the whole trip?

Source: From Balanced Mathematics Assessment for the 21st Century (p. 84), by J. L. Schwartz et al., 2000, Cam-bridge, MA: Harvard Graduate School of Education. Reprinted with permission.

1 p.m. 2 p.m. 3 p.m. 4 p.m. 5 p.m. 6 p.m.

Anne’s Ride

36

30

24

18

12

6

Mile

s tr

avel

ed

Time


Student 2: She stays in one time period, and she goes fast.Teacher: How do you know she is going fast?

Student 3: You can tell because each goes by 6. She took 6 miles to getover there at 6:00 p.m.

Teacher: What do you notice about how many miles Anne traveledbetween 3 p.m. and 4 p.m.?

Student 4: I can tell from the graph that Anne was riding at the samespeed because she was going straight up not stopping. But she wasn’t goingthe same speed because the more she goes up the more the speed she goes.

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Teacher: Why is the line straight between 3 p.m. and 4 p.m.? How far didAnne travel during that time? Why is the line straight between 4 p.m. and 5 p.m.? How far did Anne travel during that time? Why is the line straightbetween 5 p.m. and 6 p.m.? How far did Anne travel during that time?

Student 5: Because each hour she traveled 12 miles.Teacher: In what other ways can you explain Anne’s speed using the

information on this graph?

Student 6: Because the miles increase 6 miles, then 12 miles, then 18miles, then 24 each.

Teacher: What do you notice about the number of miles Anne travels andthe amount of time she has been riding?

Student 7: You look at how straight the line is and how much the mileswere for every hour and check if it’s the same as the last.

Teacher: Why does the line look straight when you compare the milestraveled and the number of hours Anne had been riding?

Student 8: Because the line that connects the dots goes up in a straightline. It doesn’t show that she stopped or went slower.

Teacher: Why does the information on the graph give this result?

Student 9: Because it is just going straight up like this (/) and not like this(/\/).

Teacher: What information on the graph makes the line stay straight?What would make it change direction?

The bicycle rides task required students to interpret a line graph in whichthe slope of the line represents speed. The process of teachers’ responding tothe initial student answers, and of the students’ then responding in writingto the teacher’s comments, cannot help but deepen and enrich the level ofstudent understanding.


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Student Writing and Special Needs

Here is a real-world mathematics problem that I experienced recently. One of the plastic spokes broke on my new umbrella. I returned it to the

store with the sales slip, which recorded the original price of $22.00 and the marked-down price of $17.60. The sales clerk ran my credit card throughthe machine and said, “Oops! I charged you another $17.60. I’ll do it againand credit you for returning two umbrellas.”

The sales clerk proceeded to credit the two umbrellas (the one I had returnedand the one for which she accidentally charged me). She entered into themachine “2 @ $17.50,” marked down from $35, as a credit of $28.

The sales clerk brought me the slip. I was expecting “2 @ $17.60” to be acredit of $35.20. What kind of math made it $28? The original umbrella was$17.60 marked down from $22, for a 20 percent markdown. $28 is 20 percentoff of $35, but the $35 was the total refund short 20 cents because the clerkhad punched in $17.50 instead of $17.60. The clerk gave up and called a man-ager, who charged me $28 to counteract the $28 returned to me. She thencredited me for two umbrellas at $17.60 each for a total of $35.20, and thetransaction was complete.

Incidentally, the first transaction in the return of the umbrella occurred at10:36 a.m. Subsequent entries were at 10:37 a.m., 10:50 a.m., and 10:52 a.m.It took 16 minutes to do the necessary mathematics so that I could be cred-ited for a returned umbrella—an inordinate amount of wait time.

This is just one small example of how each of us faces mathematics problemsevery day. The point of the umbrella story is that, in the age of computers,everyone needs to be able to think mathematically in order to do such basicthings as shop (or work in a shop). We live in a time when to hold most jobsprobably involves some technology. However, it is not enough to just push thebuttons; we also have to think about the mathematics behind our calculations.

For this reason, mathematics instruction must be customized to meet thevaried learning styles of special-needs students. Take the case of Richard, whowas identified early on in school as “developmentally delayed.” He had trou-ble with recall and with memorizing rote facts. He hated mathematics andstruggled with some basic arithmetic every school day until 6th grade.

Richard was in a self-contained special education program when he met amath teacher, Mr. D., who was excited about a standards-based mathematicsprogram for middle school about which he had recently learned. Mr. D. not

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only was impressed by this program but also believed that, with modifica-tions, Richard could succeed in using it. One of the first mathematics lessonsthat Mr. D. prepared for Richard was “The Factor Game.” Among the game’sobjectives is learning that some numbers, such as 4, have only a few factors(1, 2, 4), whereas other numbers, such as 12, have many (1, 2, 3, 4, 6, 12).

Mr. D. had already ascertained that Richard had to compensate for his lackof multiplication skills by using addition. Richard would do manipulationssuch as 9 � 8 not from memory but through repeated addition. Mr. D. wasimpressed that Richard had an alternative strategy but saw it as time consum-ing, so he had Richard write out all the factors for the numbers from 1 to 30to use in The Factor Game; Richard was also allowed to use a multiplicationchart.

After doing as Mr. D. requested, Richard was able to learn what factors are andhow to integrate them into his multiplication strategy. He learned new vocabu-lary and discovered that some numbers have more factors than others. Overtime, Richard began to recall more of the multiplication facts and was eventu-ally able to correctly answer factor-related questions on the end-of-the-year stateassessment test. At the close of the school year, Richard moved to another com-munity. Teachers, aides, and students gathered in a circle to bid him good-bye.Richard said he did not want to go to a new school because Mr. D. was the onlyteacher who had believed that he could do mathematics.

This story is touching because we know how important success is to anystudent’s academic achievement. How many Richards are there who never geta chance to succeed in mathematics? How many of our students do not reallyunderstand what they are doing and are not developing into the mathemati-cians they could become?

Traditionally, many students have been labeled as poor performers in math-ematics because they were not successful in computational arithmetic. Yetmany of these students may be conceptually capable and can be successful atalgebra, probability, and calculus. They are also different kinds of learners whohave deficits in concrete representations. Much of the mathematics that frus-trates these students is not complicated; it is just not amenable to memoriza-tion. The simple activity of scribing—be it words, numbers, or pictures—isparticularly important to the development of mathematical understanding.

Also effective for special-needs students is the MATHPLAN program (War-ren, 2002). It is informative to look at the structured writing prompts for twoof the tasks from this collection (Figures 3.10 and 3.11).

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FIGURE 3.10 MATHPLAN Task 1: “Where’s the Car?”

You and your mother go to the toy store. When you get there, you find that the parking spacesare numbered but some of the numbers have worn off the pavement. You need to help yourmother remember where the car is located, but she has parked in a spot without a number.

Source: From MATHPLAN: A Diagnostic and Prescriptive Collection for the Elementary Grades, by A. R. Warren, unpub-lished manuscript, 2002. Reprinted with permission.

1. Use this map of the parking lot. (The X on the map is your car.)• Find the number of the space that your car is parked in.• Fill in the numbers of the other spaces that are not marked.

2. At the entrance to the store, you run into a friend and his dad. After you tell them where yourcar is, your friend’s father says that they have parked three spaces away from you in the sameaisle, but he doesn’t tell you the number of their space. You want to know where they arebecause your friend has your baseball in the trunk of his car and you want to get it back.

While your parents talk to each other, you and your friend go to get the ball and put it inyour car. Using the aisle letters and space numbers, explain to your friend the fastest way(best route) to get the ball into your car and the two of you back to the store entrance.

Instructions for the Teacher

Copy the following fill-in-the-blank statements in whatever format the student is used to see-ing, such as your handwriting with large print on lined paper, worksheet-type computer-printedpages, etc. Give the student the following statements to complete:

It can’t be in space # because . We want to get to the cars as fast as

we can, so we should go to the end of Aisle that is next to space # and walk down the

aisle until we get to . We will have to to put

the ball in my car, and then we can walk back to the door.

A

Toy Store

B

C

D

1

21

31

2

12

32

3

13

24

5

25

35

6

26

36

7

17

37

X

28

38

19

10

30


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One of the most common deficits shared by special-needs students is a lackof organizational skills. In order to help them organize their thinking, andultimately their writing, a structured-response guide is of high value. As stu-dent performance improves, the guides can offer less direction and allowgreater latitude for the writing exercise.

FIGURE 3.11 MATHPLAN Task 2: Block Sets

You arrive at a part of the store where you can choose blocks of different shapes and colors tocreate your own sets. Your mother wants you to pick out sets of blocks for your three cousins,who are all about to have birthdays. She wants you to get a set of 20 blocks for Tommy andsets of 10 blocks each for Nancy and Natalie. Each set should have at least three different kindsof blocks.

These are the blocks you can choose from:

blue squares orange triangles

red rectangles green cylinders

1. How many blocks of each kind will you get for

• Tommy?

• Nancy?

• Natalie?

2. What is the total number of each kind of block that you will get?

3. After you finish gathering all of the blocks, your mother remembers that she already has agift for Tommy and asks you to put his blocks back into the bins. How many blocks of eachkind will you have after you put the blocks for Tommy back? How do you know?

Instructions for the Teacher

Copy the following fill-in-the-blank statements in whatever format the student is most used toseeing, such as your handwriting with large print on lined paper, worksheet-type computer-printed pages, etc. Give the student the following statement to complete:

I will have blue squares, red rectangles, orange triangles,

and green cylinders. I know this because .

Source: From MATHPLAN: A Diagnostic and Prescriptive Collection for the Elementary Grades, by A. R. Warren, unpub-lished manuscript, 2002. Reprinted with permission.

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Writing Supports

Students in any class are in transition. Their papers in mathematics class areworking drafts. Students who have not been able to use writing to explain theirthinking now have technological supports, in the form of voice-activationsoftware such as DragonDictate for Windows and Dragon Naturally Speaking(Software Maintenance, Inc.), ViaVoice (IBM), and Voice Express (L&H) (Edu-cational Development Corp., 2001).

Voice activation software is designed for students who have difficulty writ-ing by hand and processing text. In regular as well as special educationclasses, there are students who can say their thoughts but can’t write themwithout feeling mentally overloaded. Voice activation software strengthensthe reading and writing skills of students. Students begin by reading five orsix passages into a microphone attached to a computer, to get the computeraccustomed to their voices. If they can’t read, then the students need to lis-ten to the passages on tape and repeat them, one sentence at a time, into thecomputer. As the students say the words, the words appear on the screen.Over time, students will see sentences that don’t represent what they meantto say and will be increasingly able to self-correct.

For students who have developed the mechanics of writing but have diffi-culty with organization, software programs such as Inspiration (InspirationSoftware, Inc.) can help. Traditionally, students are taught to write first a topic sentence for their main idea, then an introduction, followed by anaccount of what happened, and then a conclusion. Many students cannothold this organizational framework in their heads; others are just anxious toget the assignment done as quickly as possible. The Inspiration programencourages students to brainstorm all their ideas, visually connect them witharrows, and add thoughts to each idea. At the end of the process, the com-puter produces an outline from which the students can develop a paper. Evenstudents for whom organization is a relatively automatic process will find thatbrainstorming and connecting ideas challenge their deepest thinking.

For example, let’s assume a student is asked to summarize what he knowsabout the sums and products of odd and even numbers, as well as to justifyhis response. The student might use Inspiration by typing his thoughts intoindividual bubbles, represented here by bullets:

• “2 + 2 = 4. An even number plus an even number equals an even num-ber. I tried this with square tiles, and every square tile could be paired withanother square tile.”

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• “2 + 3 = 5. An even number plus an odd number equals an odd number.I tried this with square tiles, and there was one tile left over.”

• “3 + 3 = 6. An odd number plus an odd number equals an even number.I tried this with square tiles, and every square tile could be paired with anothersquare tile.”

Arrows can then be drawn on the computer to connect each bubble, and anoutline can be printed from which the student may compose an answer.

One helpful resource for teachers to consider is the Technology-EnhancedLearning Environment Web site, the stated goal of which is to provide motiva-tional, cognitive, and metacognitive support to emergent as well as proficientwriters (http://ott.educ.msu.edu/newott/projects/teleweb.asp). Both high- andlow-tech solutions are available to help students organize their thoughts andtheir work by using flow charts, task analysis, webbing or networking, and out-lining. Still, teachers must be cautioned that any piece of software can beincorrectly used, leading to students’ memorizing the programs as opposed toexpressing their own thinking. It is also important to note that these techno-logical supports are relatively new. Though some of the technologies may ini-tially be difficult to use with young children, as they continue to develop andteachers use them more often, they will probably play an enhanced role in thelearning process.

Summary

Students who have opportunities, encouragement, and support for . . . writing in themathematics classes reap dual benefits: they communicate to learn mathematics,and they learn to communicate mathematically.

—National Council of Teachers of Mathematics (2000, p. 60)

Writing in mathematics helps students think. Students are confused whenthey are asked to talk about mathematics, and are afraid of giving an incor-rect answer—but as long as strict guidelines for clarity and completeness areapplied to student writing, there should be no “incorrect” answers. Writing inmathematics allows students time to wonder and to process. It encouragesthem to address the more conceptual level of mathematics, not as rote learn-ing, but as higher-level thinking.

By recording their thinking about mathematics problems, students helpexplain the solutions—and the process of arriving at a solution helps to develop


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the solution. Writing clarifies what it is the problems are asking. In order tojustify their solutions, student writers are forced to think through, and findthe meaning in, their responses.

Student writing helps teachers determine the type of learning that is occur-ring, informs them as to whether or not the students understand the lessonobjectives, and reveals the level of understanding behind the students’ algo-rithmic computations. The students’ ability to justify solutions, notice pat-terns, and draw generalizations should be evident in their writing, as shouldtheir readiness to apply lessons learned to new problems.

Writing in the mathematics classroom serves two purposes: as a mediumfor students to develop their mathematical thinking and as a guide for teach-ers to use as they assess and plan. Learning mathematics without explicit writ-ing severely limits the depth of communication that can be achieved betweenteacher and learner.

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51

Graphic Representation in the Mathematics ClassroomLoretta Heuer

What comes to mind when you see ∆? Do you see the shape literally, as atriangle? Or do you assign a symbolic meaning to it, perhaps assuming it isrelated to finding the slope of a line?

What does the word base conjure up for you when it’s used in mathemat-ics? Is your first instinct a numeric one in our familiar base-10 system, or per-haps in systems based on other numbers such as 4 or 12? Or do you think ofbase as part of a geometric figure? Did you envision the base of a polygon,which would be a line? Or of a polyhedron, which would be an area?

And what about the word or in that last sentence? Was I using the termconversationally, or mathematically? And, if mathematically, did I intend itto be the inclusive or or the exclusive or (Hersh, 1997)?

No wonder students get befuddled!The following aspects of mathematical language are particularly confusing

to students:• Technical symbols such as ∑, �, or ∆. These signs, also known as logo-

grams, stand for whole words but have no sound-symbol relationship for stu-dents to decode.

• Technical vocabulary—words such as rhombus, hypotenuse, and integer,which are rarely used in everyday conversation.

• The assignment of special definitions to familiar words such as similarand prime.

• Subtle morphology (one hundred, hundreds-place, hundredths) and the useof “little words” (prepositions, pronouns, articles, and conjunctions) in a tech-nical syntax so precise that meaning is often obscured rather than clarified.

4

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But mathematics is not only composed of words and symbols. It is also apictorial language that uses visual models to communicate. How teachers usethe language of pictures and diagrams to communicate with students andcheck for understanding is the subject of this chapter.

The following scenarios involve students using or creating graphic repre-sentations in their mathematics classrooms. The first section, Reading Graph-ics, shows students dealing with technical usage as they read graphs, charts,and diagrams. What definitions do they assign the mathematical terms thattheir teachers use? What visual model does the word conjure up for them?How can working with graphic representations provide the teacher withinsight into a student’s thought processes so that what Pimm (1987) calls“semantic contamination” can be identified and addressed?

The second section, Artful Listening in Mathematics, confronts the issue ofmathematics’ technical syntax. Here, students are drawing to learn. They arenot copying exemplars into their notebooks for later reference but rathercreating personal images of what they understand. How does syntactical sub-tlety lead them astray? How can the drawings that they create be a windowon their thinking, especially for students who cannot adequately articulatewhere their confusion lies?

Regardless of whether students are reading graphic models or creatingthem, the teacher’s questions play a key role. In addition to providing feed-back about student understanding, questions also serve a mediating function,helping students discover what they know (or don’t know) as they attempt toconstruct mathematical meaning.

In my role as a classroom-based staff developer, I observe middle schoolstudents and collect data for their teachers. After each of the lessons describedin this chapter, I met with the teacher to reflect collaboratively on what I hadnoticed. The major reflection questions that we discussed are included at theend of each case. In some situations, based on our knowledge of particularstudents, we were able to tentatively solve problems through conversation. Inall of the scenarios, the teachers and I looked for language-dependent imped-iments to learning and asked questions about how students visualized andconceptualized mathematics. These conversations heightened our apprecia-tion of how models, metaphors, and subtle phrasing can affect visual imageryand concept formation.

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53Graphic Representation in the Mathematics Classroom

Reading Graphics: Mixed Metaphors and Double-Edged Words

Getting the picture does not mean writing the formula or crunching the numbers, itmeans grasping the metaphor. —James Bullock (1994, p. 737)

As teachers, we are advised to activate students’ prior knowledge. But whatif a student’s knowledge is faulty or incomplete, or compromised by logicalinconsistencies and faulty inferences? Barnett-Clarke (2004) calls these incon-sistencies and inferences “pitfalls” and describes them as “prevalent miscon-ceptions or inaccuracies that have logical and intuitive roots and are resistantto change. . . . These inconsistencies are often so much a part of everyday usethat adults often don’t realize the potential pitfalls that lurk in how studentsinterpret what is being said or written” (p. 64). Graphic representations caninvite these types of pitfalls for students, who after all are consummate liter-alists (“pizza wedges are fractions; ergo, fractions are pizza wedges”).

In the following two scenarios, students used visual representations sug-gested by their teachers. Whereas the model used in the first scenario seemednew to the students, the one in the second scenario was firmly in place, hav-ing most likely been developed over several years of classroom exposure.

Scenario #1: Measuring Cups ActivityIn this 6th grade activity, students were asked to quadruple a cookie recipe.Although some ingredients could easily be scaled up (e.g., one egg wouldbecome four eggs), other ingredients were given in fractional or mixed num-ber amounts (e.g., 11⁄4 cups of flour). At this point in the year, the students hadnot been taught a formal algorithm for adding fractions. Prior to the activity,they had been working with fraction strips (a linear model) and pictures ofsquare pans in which “brownies” were cut into fractional pieces (an areamodel). The recipe, however, used a volume model based on standard mea-suring cups. Would the students be able to transfer understanding based onone- and two-dimensional models to this new three-dimensional situation?

For some students, the transition was not simple. When one pair had trou-ble finding an entry point into the problem, the teacher suggested that theydraw a picture. She drew a large measuring cup, which she labeled “1 cup” anda smaller one beside it labeled “1⁄4 cup.” When the teacher left their desks, thestudents discussed how they might quadruple the cups and then set to work

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drawing their picture. I visited them when they had finished and asked abouttheir sketch, shown in Figure 4.1.

When I asked the students how many cups of flour they would need for thelarger recipe, they replied, “Eight.”

Is there a logical and intuitive root to this solution? Visualize for amoment a juice glass and an iced-tea tumbler sitting next to one another on atable. How many glasses do you see? Two—one small, one large. How manycups did the students draw on their paper? Eight—four small, four large.

Questions for reflection: • In what other ways might the students have attempted to model the

problem if the teacher hadn’t offered her initial suggestion?• How might using actual three-dimensional models (i.e., a set of nested

measuring cups) before using two-dimensional representations alter the trans-fer of learning?

• What does this encounter indicate about how students understand quan-tity (of flour in the cups) and number (of separate cups)? How might this under-standing affect their grasp of fractions?

• How can teachers help students move fluidly between visual models sothat they begin to perceive mathematical generalities?

FIGURE 4.1 Measuring Cups Activity Sketch for Scenario #1

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Scenario #2: A Round Pizza in a Square Whole

“The cafeteria cook baked a pizza in a large rectangular sheet pan and put itinto the school’s refrigerator. That night, a hungry thief came by and ate halfthe pizza. The next six nights he returned, always eating half of what was left.After a week, what fraction of the pizza remained?” (Adapted from Lappan,1998, p. 46).

In the class I was observing, the teacher set out various types and sizes ofpaper for students to use in modeling a solution to this problem: grid paper,construction paper, large sheets of newsprint. Some students drew lines half-way down a sheet of paper and then began subdividing it with more lines.Others folded and refolded the paper several times. Some used scissors to cutit into a sequence of progressively smaller halves.

Not William. Although he had probably eaten school-baked pizza (whichis invariably served in rectangles) every Friday since kindergarten, he took apiece of paper and drew a large circle on it. For William, pizzas that were cutinto fractional parts had to be round. He was adamant about this, even whenthe teacher showed him that the problem described a rectangular pan.

Questions for reflection: • What does this encounter suggest about William’s understanding of

fractions? • How might language be used to bridge what William has experienced in

the cafeteria and what he has internalized from past experiences in mathe-matics class?

• How can the teacher help William move beyond a single conceptualimage and experiment with new metaphors and visual models?

• When are models useful, and when do they get in the way of new learning?

In the next two scenarios, students mapped their prior knowledge onto agiven graphic, not realizing that their personal experience was not alignedwith the model. In the first scenario, a student equated a word from his infor-mal mathematics vocabulary with formal mathematical terminology; in thesecond scenario, students used the everyday meaning of a phrase rather thanthe mathematical one.

Scenario #3: An Uphill Struggle (Slow on a Slippery Slope)

Avery was working at the whiteboard, talking about a graph he had created.He had plotted points on a coordinate grid to indicate how fast a cyclist,

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Theo, was going at various times during a trip (see Figure 4.2); for example, ifthe rider were traveling at a speed of 15 miles per hour at the 10-minute mark,Avery would indicate this by plotting the point (10, 15).

When the teacher asked questions relating to specific points on the graph,Avery was able to respond, but he became confused when the teacher askedif the cyclist’s speed had increased or decreased between two particular points.Although the teacher had emphasized in previous lessons that points neednot be connected unless there is constant change, she decided to join twopoints, (0, 0) and (10, 15), to help Avery visualize what happened during thatinterval.

To the teacher’s eyes, the line she had drawn had a steep slope, indicatingthat Theo had increased his speed from 0 miles per hour to a brisk 15 milesper hour in the first 10 minutes of the ride. Avery didn’t see it that way. Helooked at the line the teacher had drawn, tensed up, and became even moreconfused. “He kind of slows down,” he said, before changing his mind: “No!He’s both increasing and decreasing.”

FIGURE 4.2 Avery’s Cyclist Speed Chart for Scenario #3

Image/text rights unavailable for electronic book.

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The teacher continued to probe to better understand what Avery was think-ing, asking him about the 30- and 50-minute marks, where the cyclist’s speedwas also 15 miles per hour. Avery ran his hand across the line, saying, “Well,here it’s kind of leveled off. It’s flat.” Upon hearing this, the teacher realizedthat Avery saw the graph not as a representation of the cyclist’s speed, but as amodel of the physical terrain. Whereas for the teacher the steep slope indicateda fast increase in speed, for Avery it literally represented a hill, where hard ped-aling is required and speed would most certainly decrease. Avery’s misconcep-tion is not unusual: He took the phrase “a graph is a picture” quite literally tomean a picture of hills and valleys.

As teachers, we try to build on our students’ novice or intuitive vocabulary,using it as a bridge to more precise terminology. Some students understand thattheir use of informal terms during a mathematical discussion is simply a con-venient conversational substitute, lacking the precision of a more formal math-ematical term (Herbel-Eisenmann, 2002). Others, however, equate terms suchas steepness and slope, assuming that the two are mathematically synonymous.

Questions for reflection:• How might a student interpret steepness if the same data were plotted

on two graphs, with different intervals on their axes?• What are the clues that a student’s misunderstanding may be language-

dependent?• For how long, and to what depth, should a teacher continue to probe in

order to get to the logical and intuitive root of confusion?• To what extent should student-generated terminology be encouraged in

mathematical discussions?• At what point does such student-generated terminology inhibit rather

than foster understanding?

Scenario #4: Breaking Even

In common parlance, the word even suggests neatness and tidiness—an evenhemline is the same length all around. The concept of even numbers mightevoke images of animals entering Noah’s Ark two by two. But what does theterm breaking even mean?

In a 7th grade class, students were asked to interpret the line graph inFigure 4.3, which shows a bike shop’s monthly profit or loss during the courseof a year. The line zigs and zags; some months it lands above the x-axis, somemonths it settles below it, with the point for September right on the x-axis itself. For the months of April and May, profits are the same—$800. The

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question is, in which months did the shop make money, lose money, or breakeven?

Several students were convinced that because the line connecting thepoints for April and May was “flat” (i.e., parallel to the x-axis), these must bethe break-even months. The points for April and May were equidistant fromthe x-axis, just as my “even” hemline would be equidistant from the floor.Once the teacher became aware that the students’ confusion about the phrasebreaking even was causing an error, it was easy for him to explain it as the pointat which income and expenses offset each other. Note that the phrase break-ing even is not a formal mathematical term but rather a conversational idiomwith mathematical implications. Pimm (1987) calls these types of phraseslocutions—“certain whole expressions whose meanings cannot necessarily beunderstood merely by knowing the meanings of the individual words; that is,the expressions function as semantic units on their own” (p. 88).

Questions for reflection:• Did the students understand the phrase, “where income and expenses

offset each other”?• Did the teacher’s explanation seem clear to me simply because I under-

stood it from my vantage point as an adult?

FIGURE 4.3 Repair Shop Profit Chart for Scenario #4


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• Was there another double-edged word or convoluted metaphor hiddenin the teacher’s explanation that neither of us noticed, but that may surfaceat some other point?

• How can teachers become more aware of the mathematical locutionsembedded in their classroom conversations? For example, consider the manyfinancial, scientific, and statistical idioms used frequently when analyzing data.

The following scenarios explore the role of classroom climate in drawingmeaning from graphic representations. The first scenario addresses student-to-teacher communication, whereas the second listens in on students talkingto each other.

Scenario #5: Envisioning the Invisible

One group of 6th grade students I observed had just finished a unit on frac-tional concepts. Before their teacher introduced them to formal operationswith fractions, she had them work in a probability module that required theuse of fractional notation, equivalent fractions, and the informal addition offractions. On this particular occasion, the students were using a spinner sim-ilar to the one shown in Figure 4.4, portioned into wedges of three differentdesigns.

A single one-sixth portion of thespinner has stripes; two noncontigu-ous sixths have dots. The remainingsixths have hearts on them: a one-sixth slice on the right side of thespinner and a one-third wedge onthe left. The students’ task wastwofold: First they needed to spinthe spinner 30 times and report, infractional form, the results of land-ing on each design. Second, theywere to analyze the spinner, bymeasuring it with an angle ruler orby some other method, to determinethe fractional portion of stripes,dots, and hearts.

Most of the students opted not touse the angle ruler. Some cut out thespinner, creased it across the middle,

FIGURE 4.4Spinner forScenario #5


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and then folded the resulting semi-circle into thirds using the one-sixth mark-ings as a guide. Others cut the spinner apart and repositioned the pieces so thatthose with the same design were adjacent to each other. Still other studentsdrew what they termed “invisible lines” dividing the one-third chunk in halfso that the spinner was composed of six one-sixth sections.

The class discussion that followed this activity was lively. Students sharedtheir strategies, argued as to whether or not the sections with the same designmust be adjacent, suggested which designs one should “bet on” to win, notedhow a “50-50 chance” really meant a 50 percent chance, and discussed the dif-ferences between their experimental results and what each fractional portionof the spinner predicted. It looked as if the class had a solid understanding ofthe concepts.

As the discussion wound down, Sonya spoke for the first time. Pointing tothe line splitting the one-third wedge in two, she asked, “Will they still besixths if the invisible line isn’t there?”

Questions for reflection:• Did Sonya’s question indicate confusion about equivalent fractions? Or

was she trying to clarify a precise perception of the visual image? • What understanding did Sonya have about the concepts her classmates

had just discussed? • Could Sonya have asked her question earlier? Or did her classmates’ vig-

orous discussion help her formulate her question?• What classroom dynamics were in place that allowed Sonya to feel com-

fortable asking such a basic question?

Scenario #6: The Wordsmiths

Students often know that something is happening in a table or graph, that apattern is right there. They see it clearly, but. . . . It’s frustrating for any of usto grope for words.

In an 8th grade class that was starting a unit on exponential decay, the stu-dents were visibly engaged as they pointed at numbers on the table of valuesthey had created. Their hands were flailing, they were rolling their eyesupward, searching for a word somewhere in the back of their minds and,when not finding one, inventing terms as they tried to express what theywanted to communicate. In short, they acted precisely the way travelersabroad do when forced to communicate in a language not their own.

Lacking precise terminology, these students began to describe what theysaw happening on their table of values by relating decay to exponential

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growth. If things grew exponentially by doubling in yesterday’s lesson onexponential growth, then today things seemed to be getting smaller by“undoubling.” Or maybe it is called “doubling down.” It certainly wouldwork if you “divided it in half.” Would that be the same as “taking half of it”?Or “dividing by two”? Would that be “halving”? Is that “the opposite of dou-bling”? Nothing was said about reciprocals or inverse operations . . . yet.

Questions for reflection:• What might the teacher have done to foster a classroom climate in

which students felt comfortable talking to each other about emerging math-ematical ideas?

• Would this particular mathematical conversation have occurred if thestudents didn’t have a graphic “prop”?

• How did the visual representation act as a catalyst for student discourse?• How might conversations about a graphic display encourage students to

put their mathematical perceptions into writing?• Now that a conversation had begun on the topic of exponential decay,

how might the teacher build on the richly descriptive terms that the studentscreated to press for more precise mathematical terminology?

Artful Listening in Mathematics:The Subtleties of Syntax

“Show and Tell” is one of the earliest ways that children learn to communicatein the classroom. Youngsters who are normally shy and inarticulate whenasked to talk in front of a group start to relax when holding a prop that theycan talk about. What can this primary-grade communication strategy suggestto teachers who are trying to help older students develop oral communicationskills in mathematics? Moreover, how might teachers use this approach todraw out misconceptions and flaws in student thinking? The answer may be assimple as asking students to create a “prop” that they can talk about by liter-ally drawing one out. If a teacher suspects that a student’s concept image is“sketchy,” wouldn’t it be useful to get that image on paper? When students cre-ate visual images, they are externalizing their mathematical thought processes,even if they do not have the precise mathematical language with which toexpress their ideas.

In the following scenarios, students were deep in the process of construct-ing mathematical meaning. However, unlike the students in the preceding

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section, who often seemed frustrated or confused, these students were confi-dent that they had listened to their teachers and understood what had beensaid. All these students, however, missed crucial nuances in their teachers’mathematical syntax and unfortunately began to build conceptual misunder-standings based on faulty assumptions. Because these students had a less-developed mathematical grammar and syntax, they struggled to translate theirmathematical ideas into words.

Students are often asked to look at a mathematical diagram and interpretit orally. In the following scenarios, the students did the opposite: having lis-tened to mathematical words, they created interpretive sketches. As you readthese scenarios, try to visualize the students’ drawings as you consider the fol-lowing questions:

• What did the teacher mean?• What did the teacher say? • What did the student hear? • What visual image did the student create to match what he or she heard?

Scenario #7: Where’s the Fourth Fourth?

Rather than introduce fractional concepts using her own visual models, one6th grade teacher routinely encouraged her students to draw what theythought she had said about fractions during a lesson. In this fraction module’sfirst activity, students were given 1" � 81⁄2" strips of blank paper that theyattempted to fold into halves, thirds, fourths, fifths, sixths, eighths, ninths,tenths, and twelfths. The class worked with their fraction strips for severaldays, naming them, comparing them, finding fractions that were equivalentto each other, and, finally, labeling the strips in symbolic a/b form. One stu-dent, Benjamin, showed me his fraction strip for fourths. It was preciselylabeled on each of its three folds: 1⁄4, 2⁄4, 3⁄4 (see Figure 4.5).

When I asked him, Benjamin told me that this strip represented thirds:“The whole thing is divided into three parts.” When I asked about some ofthe other strips, each of which was correctly labeled, Benjamin followed thesame line of thinking: his name for the strip was based on the last numeratorin the series. When I asked him about the strip that was folded in half, heseemed confused. Intuitively, he knew that he should call it a “halves” strip,but the numerator was a “1.” The system he had carefully constructed didn’twork in this particular case. How can that be?

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Questions for reflection:• What distinction was Benjamin making between the terms thirds, three

folds, and dividing into three parts? • How might students conceptualize fourths differently, depending on

whether they are asked to label the strip’s segments (1⁄4, 2⁄4, 3⁄4, 4⁄4) or its folds(1⁄4, 2⁄4, 3⁄4)?

• What consideration might the teacher need to give to precision of lan-guage when providing directions for this task?

• What role might peer-to-peer discourse have played in helping Ben-jamin test his conjecture?

• What questions might a teacher ask to check more deeply for understand-ing if, at first glance, a student’s thoughtfully done work is apparently correct?

Scenario #8: Three Three/Ten Combos

Later in the same module, Benjamin’s class was introduced to area models forfractions. The first activity for this unit had students divide pictures of squarebaking pans into 15 equal-sized brownies. Although a few students “cut” thepan into thin, biscotti-like brownies (a 1 � 15 arrangement), most drew linescreating a 3 � 5 grid. When asked to cut the brownies into 30ths, all but onestudent decided that 30 ultra-thin biscotti (1 � 30) were too tiresome andimprecise to draw. The most popular arrangements were 5 � 6, 3 � 10, and 2 � 15 (the 1 � 15 biscotti cut crosswise into halves).

FIGURE 4.5 Benjamin’s Fraction Strips for Scenario #7

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This model of vertical and horizontal cuts became familiar to the studentsafter several days. Benjamin was building on this understanding of area mod-els for fractions when he started to work with decimals. After an initial activ-ity in which the students planned a garden using a hundredths grid, theteacher distributed a paper on which there was a large square divided verti-cally into tenths. When asked to model three-tenths, Benjamin immediatelytook up his pencil and divided the square crosswise into thirds, creating 30equal parts as shown in Figure 4.6.

When questioned about his approach, Benjamin responded, “This is three”—indicating his horizontal cuts—“and tenths”—indicating the vertical linesprinted on the template.

Questions for reflection:• From what assumptions might Benjamin have been working when he

approached this new concept?

FIGURE 4.6 Benjamin’s Tenths Grid for Scenario #8

Tenths

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• Where was there a language-based component in his confusion? • What do the phrases “three-tenths,” “three and tenths,” and “thirds and

tenths” mean mathematically? What do they seem to mean to Benjamin?• In a roomful of 20 students, how difficult is it to hear the subtleties of

different word forms?• What might Benjamin’s drawing imply about what he heard?• How could the teacher use Benjamin’s drawing to encourage him to

express his personal understanding in words?• How might making additional sketches have helped Benjamin commu-

nicate what he heard and understood?

Scenario #9: Holes in Her Logic

Sarah’s 6th grade class was finishing a unit on fraction-decimal equivalents.On this particular day, the teacher gave the students a list of foods donated to a hurricane relief project and asked them to determine how the food couldbe equally distributed among 24 families.

Sarah began by drawing 24 squares to represent a box for each family. For48 cans of cocoa, she counted aloud as she put one dot representing one caninto each box. She repeated this procedure once more so that each squarecontained two dots. Then she erased the dots. She used the same strategy forthe 72 boxes of powdered milk. When she got to the 264 juice boxes, how-ever, she switched to a calculator. She entered the division problem correctlyand wrote the result, 11, on her answer sheet. When faced with the task ofdividing up 12 pounds of cheddar cheese, Sarah again reached for her calcu-lator and pressed the correct keys: 12, ÷, 24, =. Rather than writing the answeron her worksheet, she reached for the sheet of box diagrams and began put-ting little circles in each of the 24 boxes, as shown in Figure 4.7.

What was Sarah doing? Had she forgotten to record her answer and movedinstead to the next question, which involved dividing up six pounds of Swisscheese? Is that what she was drawing?

Sarah seemed proud and confident when she explained her work to me. Iasked her where the five dots that she had put into each box had come from.Why, she had read that right off her calculator screen: they were “point-five.”

Questions for reflection:• What windows to Sarah’s thinking do her cocoa, milk, and cheese draw-

ings provide?• How did Sarah seem to understand division in general? What about divi-

sion in situations in which the quotient is greater than one? When the quo-tient is less than one?

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• What did Sarah believe about the decimal point?• What was Sarah hearing, and how did this influence the manner in which

she conceptualized decimal fractions?• What is appropriate calculator use for students beginning to work with

fractions and decimals?• Would language-based cues, such as the teacher’s asking how to divide a

12-pound wheel (or chunk) of cheese among 24 families, suggest useful visualmodels? Or might metaphors create further confusion?

• What might the teacher do to reinforce the use of proper terminologywhen students are working with decimal fractions?

Scenario #10: What Does More Mean?

One way that teachers can use language to help students make sense of math-ematics is to contextualize problems by “wrapping words around the num-bers.” The words, in turn, may give rise to images that students can use whenformulating solutions. However, paragraph-sized narratives present problemsof their own. Often numbers are written as words rather than as numerals(e.g., “five” rather than “5”). There might be extraneous information thatneeds to be disregarded. Or maybe the question that students need to answeris at the end of the paragraph, and they’ve forgotten the details they’ve justread. It is always useful for teachers to assume their students’ mind-set and

FIGURE 4.7 Box Diagrams for Scenario #9

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attempt problems from their perspective. What phrases do you think you’dwant clarified? What language would you need to have “debugged”?

Consider the following problem adapted from “Comparing and Scaling,”a 7th grade Connected Mathematics module on proportional thinking (Lappan,2002b). It may help you to visualize the problem if you pick up a pencil andmake a drawing of your own.

You and 17 colleagues are meeting for dinner at Nona’s Italian Restaurantbefore a parent open house. Because you want to socialize as a group, youprefer not to sit at one long banquet table, nor do you want to be split upamong many small tables. The manager accommodates by pushing togetherthree small tables to create seating for eight, and four small tables to createseating for ten. When dinner is served, one pizza is placed on each of thesesmall tables: three for the group of eight, four for the group of ten. In eachgroup the pizza is shared equally among tablemates. Would you get thesame amount of pizza regardless of where you sat? If the amounts were dif-ferent, at which table would you get more? Explain your reasoning. (p. 30)

Before reading the classroom description, ask yourself the following questions:• As written in the students’ textbooks, the problem has pizza being served

at cafeteria tables. In the revised version above, the problem is situated in anadult context. Did the restaurant context help you develop a mental image ofthe mathematics involved?

• Did you create a sketch when solving the problem? If you didn’t, why not?If you did, was it helpful in solving the problem? Were your eyes drawn back toread the text as your hand created the visual image? Did drawing help you clar-ify what you had read? Did your experience in sketching to solve the problemmirror the opinion of Reehm and Long (1996)—namely, that drawing helps us“to perform the indicated computations, answer questions, or draw appropriategraphs or diagrams . . . [and] is essential for comprehension” (p. 38)?

The lesson I observed started with the teacher’s referring back to prior workwith rates, ratios, and proportions. After receiving student feedback about thesetopics, the teacher read the day’s problem to the class and showed them atransparency of the seating arrangement. Armand’s 7th grade textbook alsohad a diagram of the problem’s seating arrangement, and he began by copyingit. Working alone, he sketched two rectangles to designate tables but did notdraw anything to indicate seats or people around them. He continued by draw-ing circles on each rectangle to represent the appropriate number of pizzas. He

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then sectioned each circle into either eight or ten wedges, depending on howmany people were to be seated at each table. (Creating tenths was bothersomeand involved multiple erasures.) Then he stopped. Other students continuedto work, setting up proportions, changing 4⁄10 to the equivalent fraction 2⁄5, andfinding the decimal equivalents for 4⁄10 and 3⁄8.

Armand’s hand was the first one up when the time came for students toreport their findings. He used his diagram to show that each person at the tableof eight will get three pieces of pizza, while each person at the table of 10 willget four pieces. He reasoned that a person will get more at the table of 10because, “You get more pieces.” The teacher pressed for further explanation:

Teacher: But I’m not asking you how many pieces. How do youknow it’s more pizza? Armand: Well, I cut each pizza in 10.Teacher: But what about the size of each piece of pizza?Armand: I have one piece from each pizza for each person.Teacher: But why is that more pizza?Armand: Because it’s four pieces each.Teacher: But why is that more pizza?Armand: Because it’s four!

Questions for reflection:• Compare the use of the singular noun pizza in these two sentences from

the problem. How is the meaning of the word pizza different in each instance? – “Each pizza is cut into eight slices.” – “In each group the pizza is shared equally.”

• Read the following paragraph; then think about how the word more isused in everyday conversation and how it is used in a mathematical context:

The distinction between count nouns and mass nouns . . . is the distinctionbetween discrete and continuous quantity. When we talk of discrete quan-tity, we use count nouns and we may have many or few of them. On theother hand, when we talk of continuous quantity, we use mass nouns, andwe may have much or little of the referent entity. In comparing discretequantities we talk about more and fewer, while in comparing continuousquantities we use more and less. (Schwartz, 1996, p. 8)

• Refer back to the text of the Nona’s Restaurant problem, and analyze itfor instances of discrete versus continuous quantity. Where do you note ashift from one concept to the other?

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• How might the teacher explain to Armand the distinction between“how much pizza” and “how many pieces”?

• How could the teacher use language in conjunction with Armand’sdrawing to help him reframe the problem?

• At the quick-checkout register of your supermarket, does the sign say“Ten Items or Less” or “Ten Items or Fewer”? What is the semantic difference?

When teachers are asked in post-observation conferences to reflect on howstudent drawings can inform their practice, three themes emerge. Teachersfeel that the drawings

1. Make the students more aware that they’re “speaking mathematics” inclass,

2. Show a need for greater precision in the students’ use of mathematicallanguage, and

3. Suggest areas in which directions and explanations should be moreclearly phrased.

Suggestions from Teachers

How can we help students understand the graphic and figurative language ofwhat Krussel (1998) terms “the mathematical community’s existing body ofmetaphors” (p. 49)? Many suggestions emerged during my conversations withteachers:

• Combine the verbal with the visual. When dealing with words that haveambiguous meanings, consider having students fill in graphic organizers orconcept maps.

• Monitor your use of metaphors, models, and idioms. Experienced mid-dle school teachers routinely monitor their vocabulary in order to avoid usingeveryday words that have titillating double meanings for students. Similarly,teachers should identify terms that have both conversational and separatemathematical meanings, clarify these differences for students, and watch forany incorrect but logical and intuitive roots that hinder understanding.

• Assume positive intent. Students may couch their mathematical ques-tions or observations in offhand, joking, or even dismissive tones. Speak-ing half-formed thoughts in front of their peers can be so uncomfortable thatstudents may feel safer playing the clown than risk asking what they fearmight be a foolish question. As teachers, we need to believe that in everyquestion asked, in every answer given, there is the kernel of a student’s strug-gle to understand and to make mathematical meaning.

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• Once students are comfortable showing their understanding by drawingpictures, encourage them to move to a more stylized representation of thesame concept. Moving from a detailed picture to a simplified shape, oftencalled an icon, is another step toward mathematical abstraction.

• Consider the order in which individual students select different forms ofrepresentation. What sequence seems to promote depth of mathematicalthinking for the specific topic being studied? What implication does this havefor lesson planning?

• Actively point out connections to other mathematical representations.Clement (2004) describes a representations model shaped like a five-pointedstar in which the contextual, spoken, pictorial, tactile, and written symbolsare interconnected and where the goal is for students to become fluent intranslating one representation into another.

To help students better visualize the language of mathematics, teachers sug-gest the following practices, which can be incorporated into classrooms withminimum disruption:

• Articulate and enunciate, being especially conscious of word endings,pronoun referents, and “little words” that tend to be murmured or mumbled.

• Buy pencils, and keep them sharpened. Sharpened no. 2 pencils andgood erasers need to be classroom basics if students are expected to commu-nicate by drawing.

• Provide enough time for students who need to draw in order to learn.The kinesthetic act of putting ideas on paper, whether by drawing or writing,takes more time than talking. It is reasonable to assume that one benefit ofdrawing and writing in mathematics is that it slows down our thinking.

• When assigning open-response questions that require both writing anddrawing, suggest that students create the diagram first in order to have some-thing concrete to write about.

• Make copies of key textbook pages so that students can write notes onthem, add diagrams, doodle in the margins, and underline words. This allowsstudents to engage with the text in a tactile, kinesthetic, physical way.

• Have students use a graphic organizer, preferably one that incorporatesboth words and images. According to one study, students who were taughthow to use graphic organizers for test preparation retained significantly moreinformation than did other students (Dickson, Simmons, & Kameenui, 1995).The Verbal and Visual Word Association (Readence, Bean, & Baldwin, 1998) isa powerful but simple graphic organizer consisting of a plain index card

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divided into four quadrants with places for a word or phrase (upper left), itsdefinition (lower left), a visual representation (upper right), and a personalassociation (lower right).

• Designate an area for sketching when creating templates and work-sheets. Students who find it useful to scribble and sketch often do so on apiece of scrap paper that gets tossed in the trash at the end of class. Allottinga section of the page for drawing indicates that you value and expect to seevisual thinking.

Mathematics: A Visual Language for All Students

Although student-generated drawings can help teachers root out mathemati-cal misconceptions, they are much more than a diagnostic tool. For kines-thetic learners and geometric thinkers, sketching may be a preferred strategyfor processing the language of mathematics into learning. For these students,drawing is not just a way to make their thoughts about mathematics visibleto others; it is a device to capture the language of mathematics in order tomake it visible to themselves. Even for students who approach learningthrough other pathways, the kinesthetic act of drawing may be valuable.Drawing slows students down and allows them to self-correct their thoughtswhile their hands are sketching; it also helps them to keep track of and recordtheir solutions (Albert, 2000).

Research studies have shown that drawing helps students better under-stand the material. As Borasi, Siegal, Fonzi, and Smith (1998) note, “The actof recasting meanings generated in one sign system (language), into another(visual art), is intended to invite reflection from a different perspective, amove that can lead to new insights. Because no code for translating languageinto visual images exists prior to the creation of sketches, students mustinvent their own. It is this act of crossing the gap between alternative sym-bolic systems that gives sketching its generative potential” (p. 280).

The classroom scenarios presented in this chapter suggest that mathemati-cal language influences the drawings students create, first in their thoughtsand then on paper. But might students’ visual images also influence theirmathematical language? It seems probable that when drawing accompaniestalking and writing, students develop both visual and verbal literacy.

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72

Students construct meaning of the mathematics they encounter throughmany experiences. They travel through their daily lives bumping up againstmathematics without even knowing it as they play with toys and games orwork with money and tools. As they progress through our mathematics classes,they adapt to a dozen or so different teachers and a variety of textbooks.

As teachers, we naturally hope our students will attain deep mathematicalunderstanding in our classrooms. One powerful tool for enhancing thatunderstanding is classroom discourse. The NCTM’s Principles and Standards forSchool Mathematics (2000) speaks to the need for students to make conjec-tures, experiment with problem-solving strategies, argue about mathematics,and justify their thinking. There is general agreement that discussion andargumentation improve conceptual understanding.

For the purposes of this chapter, I am defining discourse as the genuinesharing of ideas among participants in a mathematics lesson, including bothtalking and active listening. Such sharing occurs on many levels—betweenteacher and student, between student and student, within small groups, andwithin the whole-class group. I will focus mainly on whole-class discussions,keeping in mind that stirring up an argument can lead to improved learning.Great conversations happen only when genuine discourse becomes the cul-ture of the classroom.

What Discourse Looks Like As a math teacher coach, I have either observed or been a participant in count-less classroom conversations, ranging from traditional teacher-centered inter-changes to discourse-rich student-led discussions.

Discourse in the Mathematics ClassroomEuthecia Hancewicz

5

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73Discourse in the Mathematics Classroom

• Traditional. The teacher and several students talk. The teacher asks mostof the questions, which are directed at specific students who in turn responddirectly to the teacher. There are brief conversations between the teacher andindividuals. Other students are expected to listen. When a student doesn’trespond to a question as the teacher expects, prompts from the teacher gen-erally lead to a preplanned answer.

• Probing. The teacher is still the leader, and conversations are still betweenthe teacher and individual students, but questions are more open. They stemfrom the teacher’s desire to hear about students’ thinking, rather than from aneed to move students along a planned route—from a desire to elicit ideas forfurther thinking and to pique the interest of other students. Common ques-tions in this type of interchange include, “How does that fit with what [anotherstudent] said?” and “How did you decide to do what you did?”

• Discourse-Rich. Young people work toward mathematical understandingby sharing ideas with each other and the teacher. As Stigler and Hiebert (1999)note, “Students must . . . learn to question and probe one another’s thinkingin order to clarify underdeveloped ideas” (pp. 90–91). When this type of con-versation becomes the culture of a classroom, students uncover new ways toconceptualize mathematics.

One day, Ariel, a student in Mr. Ryan’s 7th grade class, was explaining hismethod of estimating the distance around a trapezoid drawn on unit gridpaper so that AB = one unit (see Figure 5.1). “I counted the units and theperimeter is five units,” he said, demonstrating on the projected image of theshape by counting BC as two units and each of the other sides, AB, AD, andDC, as one unit.

Belinda’s hand shot up. “I thinkit’s more than five units,” she said.Several other student looked per-plexed, so Mr. Ryan gave the classone of his well-known “I wonder”looks. Ariel was at the whiteboard, sohe explained his thinking again, butBelinda wasn’t convinced. “Thenyou come up and do it!” said Ariel.

At this point, Mr. Ryan walkedto the back of the classroom andsat at an empty student desk.

FIGURE 5.1 Ariel’s Trapezoid

A

B

D

C

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Belinda said she thought the segment CD was longer than one unit becauseit was tipped. There was some general chatter among the classmates; then Char-lie said, “I agree, and I can prove it.” He took a ruler to the front of the class-room, laid it along the AB segment and put a finger at the spot where one unithit the ruler. The class attentively listened as he explained, “I call this one uniteven though it isn’t matching one inch on the ruler.” Then he laid the ruleralong the DC segment, thinking he’d show that it was longer than one unit.

His classmates weren’t satisfied. Several accused him of making up the unitand moving his finger.

Now Dara jumped up. “I can prove it!” she said. Her method involvedmarking the length of the AB segment on a piece of paper and comparing thatto the length of the DC line. Her demonstration convinced the class that theDC line was longer than AB, so the perimeter had to exceed five units.

Mr. Ryan returned to his role as leader, happy to have let the students con-duct this investigation on their own.

When the classroom climate fosters genuine student discourse, studentsreact to their classmates’ ideas, asking questions and checking for understand-ing. Conversations continue without the need for teacher participation orsupervision.

Creating Discourse-Friendly Classrooms

Moving from traditional student-teacher interchanges to rich classroom dis-course is not an easy or superficial change; it requires a major shift in one’sdefinition of effective teaching. If we believe that the goal of teaching is forstudents to become proficient at a set of procedures, then it is logical to craftlessons that help them move gradually along a planned sequence of increas-ingly more sophisticated procedures. However, if we believe that the goal isfor students to build mathematical understanding as well as efficient proce-dures, a well-crafted lesson depends upon the students effectively sharingtheir strategies and ideas. Thoughtful teachers use student ideas as spring-boards for powerful lessons. Discourse is more than a teaching technique; itis a framework on which to build effective mathematics lessons.

In an article entitled “Talking Their Way to the Middle of All Numbers,”Kristine Woleck (2001) gives us a view of how she used student conversationsto build a 1st grade lesson about infinity. In the midst of a conversation abouttemperatures, one of Woleck’s students commented that 25 degrees was“halfway between 20 and 30! It’s in the middle.” Another student’s response

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to this observation was, “I’m wondering, what is the middle of all numbers?”Woleck goes on to tell us how she was able to let this kernel of an idea growinto an exploration of the abstract concept of infinity. The discussion wasaccessible and meaningful to all the students regardless of their level of math-ematical understanding. Student voices give a sense of the flow of their ideas:

Ryan: “Fifty is the middle, because 50 plus 50 equals 100.”Elizabeth: “But you can count to more than 100.”Antoine: “There’s no middle number because numbers never stop.”Whitney: “If you count to an even number, then there will betwo numbers in the middle. If it’s an odd number, then there’s onenumber in the middle.”Kevin: “The problem is numbers never stop.”Whitney: “Well, the numbers go so far up, but what’s the biggestnegative number?”

Woleck had created a classroom environment in which students sharedtheir ideas and challenged each other under her guidance. As she notes in herarticle, her “ears were open that day” and she was able to let the studentvoices lead an exploration of one of mathematics’ big ideas (p. 30).

As I coach teachers, I help them plan lessons, I observe the lessons, and Ifollow up with reflective conversations. I’ve been part of teachers’ lives asthey work to shift from traditional teacher-centered classrooms to discourse-rich ones. This difficult switch requires a whole new belief about what learn-ing is. I am convinced that small changes can help teachers move forwardwithout a major upheaval. The following is a list of personal suggestions forsmall but significant changes teachers can make:

• Arrange furniture so that students can easily turn to see each other. Theymust be able to speak and listen to classmates.

• Encourage students to direct questions and explanations to the class,rather than to the teacher.

• When recording ideas on chalkboard or chart paper, use the students’words as much as possible. This is a matter of respecting their ideas.

• Try not to repeat or paraphrase everything students say. Paraphrasingcan give the impression that the student is being corrected and may indicateto others that they don’t need to listen unless the teacher speaks.

• Remind students that conversation is a two-way operation requiringboth talking and listening.

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• Stand in a variety of spots. As students turn to look at you, their viewsof the classroom and their positions relative to classmates will shift.

• Remember, students listen harder when a peer speaks than when an adultdoes!

• Give students time to think. Wait time or brief writing moments helpstudents to solidify ideas and formulate good questions.

• Arrange lessons so that students have a product to share as they explaintheir thinking. They might illustrate ideas on chart paper or overhead trans-parencies or demonstrate by using manipulative materials.

Another strategy is to use favorite activities to foster discourse. In her bookTeaching Mathematics Vocabulary in Context (2004), Miki Murray provides direc-tions for the game, “I Have, Who Has?” and explains how her class expandedon it (Figure 5.2 shows a sample set of cards for use in the game):

The game (exercise, really) goes like this: The cards are randomly distrib-uted among students. Everyone gets at least one card, but many studentswill have more than one. A student is selected at random to go first. Sheselects one of her cards, ignores the “I have” at the top, and carefully readsthe “Who has” definition. Students must focus and listen carefully todecide whether the definition matches one of their words. The person whohas the word being defined calls out “I have” followed by the word. Thatperson then reads the “Who has” definition on her card. The game contin-ues until all the definitions have been read. The cards are designed so thatthe student who began the game will have the final word at the top of thebeginning card. (p. 63)

The love Murray’s students had for this activity led them to create a versionusing their own terms and definitions. This led to all sorts of student conver-sations: brainstorming words to use, figuring out the criteria for high-qualitydefinitions, peer critiques, and testing the game.

As teachers shift toward increasingly discourse-friendly classrooms, we mustcede some control. Managing conversations among students about theiremerging knowledge allows discussions to move beyond the planned lesson.Teachers must understand mathematics at a deep level in order to followyoung people’s unconventional reasoning. We must understand the conceptsthat provide the foundation for each day’s learning, recognize how today’s les-son builds toward future concepts, be clear about connections across strands,

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and keep sight of the overarching ideas of mathematics. As Woleck (2001)reminds us, “the ‘big ideas’ of mathematics develop at different rates for eachlearner. . . . Children will revisit and refine their understanding” (p. 31).

Thoughtful participation in discourse with students informs teachers asthey make instructional decisions. Here are several ways to let student ideastake the lead in class:

• Involve students in engaging and challenging problems.

FIGURE 5.2 Sample Cards for the “I Have, Who Has?” Game

I have RHOMBUS

Who has a quadrilateralwith only two parallel sides?

I have TRAPEZOID

Who has a chord of a circlewhich goes through thecenter?

I have DIAMETER

Who has a nine-sidedfigure?

I have NONAGON

Who has figures withexactly the same shape andsame size?

I have CONGRUENT

Who has an angle withexactly 90 degrees?

I have ACUTE TRIANGLE

Who has a rectangle withfour congruent sides?

I have SQUARE

Who has a polygon with sixsides?

I have HEXAGON

Who has lines in a planethat never intersect?

I have PARALLEL LINES

Who has an angle withexactly 180 degrees?

I have STRAIGHT ANGLE

Who has a parallelogramwith all sides congruent?

I have RIGHT ANGLE

Who has a five-sidedpolygon?

I have PENTAGON

Who has the value of pi tothe hundredths place?

I have 3.14

Who has an eight-sidedpolygon?

I have OCTAGON

Who has the instrumentused to construct circles?

I have COMPASS

Who has the type of trianglewith all angles less than 90degrees?

Source: Adapted from “I Have, Who Has?” by the Association of Teachers of Mathematics in Maine, Winter 2000,ATOMIM Newsletter, p. 11. Copyright © 2000 by the Association of Teachers of Mathematics in Maine. Used withpermission.

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• Ask open questions to stimulate student thinking. (Examples: “Whatdoes this make you wonder about?” “Are there patterns?” “Is this logical?”“Can we estimate a solution?”)

• Listen carefully to student responses. Ask for clarification so that youand others really understand. Diagrams, physical models, and nonmathemat-ical vocabulary may help students explain.

• Train students to listen to their classmates’ observations by asking ques-tions that engage: “Does this work?” “What do others suggest?” In one class-room, the teacher routinely used a single word to stimulate. He would simplysay, “More?” and students would add their ideas to the interchange. He wouldrepeat the word several times with no further comment until no one had any-thing else to offer. Students would then discuss the lesson, uncovering andsorting its key concepts.

• Honor diverse ideas, methods, and examples from varied sources. Stu-dents may explain most clearly by using pictures, computer illustrations, andstories.

• Honor ideas even if they’re incorrect. Do not quickly agree or disagree.Students will come to realize that you are giving them time to think and tojustify. Often, as students explain erroneous thinking, they uncover their ownerrors or classmates step in to clarify or correct them.

• Encourage mathematical arguments between students. Kids love it! Thewhole class benefits when students follow along as a few classmates arguetheir way through mathematical misunderstandings.

• Remember, confusion is okay. Years ago, my insightful mentor called it“disequilibrium.” Some of the best learning happens when we sort out whatit is that has pushed us a bit out of balance. Be sure students know you aredeliberately letting them be confused and that this is based upon your knowl-edge of how people learn, as this tactic may not match what previous teach-ers have done.

• Take time to let students share different problem-solving methods. Evenwhen a correct solution has been shown, ask if there are other ways to do theproblem. This helps to deepen understanding and makes students more will-ing to work with their own strategies, rather than thinking there is only onecorrect method.

• Tangents are good. If an idea emerges from class discourse and capturesthe group’s imagination, go with it. Keep the big ideas in mind, and don’t for-get to return to the concepts you plan to teach, but capture those teachablemoments.

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A few years ago, my 8th grade class and I were examining the area of par-allelograms. My students were having a very hard time moving to the gener-alization that an area could be calculated by multiplying the base length bythe height (and not by the side length). One student tried to show us herunderstanding by drawing a rectangle with “hinged” corners. Her drawingwas confusing, so she asked if we could make some “hinged” rectangles.Another student suggested that we could make all kinds of shapes with“hinged” corners. My thought was, So much for today’s lesson!

Our detour into this investigation took an extra day, with much design andconstruction work being done at home, but the rich mathematical under-standing that evolved was noteworthy. Students really learned what happensto the area inside a polygon as side lengths remain stable while altitudesvary—and they discovered special characteristics of triangles as a bonus!

• Decide how much leadership your students need. I’ve implied that weshould let students’ ideas lead, but this doesn’t mean that the class just moveswithout teacher direction. We make dozens of decisions in the course of eachclass period. It’s up to us to ground lessons in the important ideas. Studentsneed leadership; they need information, clarification, modeling, and supportof many kinds. As we work to let student ideas lead, our skills as traditionalteachers are still vitally important.

Pisauro (2002) provides some excellent instructional tips for facilitatingclassroom discussions. They may be summarized as follows:

• Focus student attention on a problem, puzzle, figure, process, question, orset of numbers. Stimulate discussion by asking the following types of questions:

– What do you notice about . . . ?– Do you see any patterns?– What is similar about and ?– What is different about and ?– How do you think this works?– Why does this work/look this way/give this result?– What questions do you have?– What can we do with this information?– What do you want to know?

• As students respond, list their statements on the board where all can see.Encourage them to write the ideas in their mathematics notebooks. Ratherthan rephrasing their responses, ask, “How do you want to say this?”

• When observations or questions are brought up by one student, ask,“What do the rest of you think about this idea? Does it make sense?” Encourage

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them to consider other examples that would show that the observation is or isnot always true.

• Motivate students to search for patterns, delve deeper, and generalize.• If students are making mistakes or doing something awkwardly, ask

them, “Is there an easier or more efficient way?” or “In what other ways couldthis be done?” rather than telling them how to do it.

• If students have difficulty thinking about a concept, suggest examples toconsider or play devil’s advocate. Ask “What if?” questions.

• Counter questions with questions instead of explanations. People tendto blank out when one person asks a question and the teacher immediatelygives an explanation. Asking another question or saying, “What do the rest ofyou think about that?” tends to engage everyone’s thinking.

• Even when a solution is successful, take time to ask whether anyone didthe problem a different way or discarded an idea. Help students to build con-fidence in their own ideas, knowledge, and insights by showing that problemscan be solved in a variety of ways.

Discourse and Computation

We lament that students can’t do arithmetic computation. Though we hate tospend precious time on drills and practice, we know that efficiency comesfrom experience. But there will be less need for drills if we foster rich discourseas we help students develop efficient algorithms. The following case illus-trates the richness we can provide when we take advantage of an opportunityfor real mathematical conversation.

Kerri, a 5th grade teacher, wanted her students to study two-digit multipli-cation. Students used base-10 blocks to figure out several products, startingwith 2 � 14. After they did this a few times, Kerri asked them to look at thearithmetic example she wrote on the chalkboard:

14� 2

She turned to the class and asked, “How should we start?” Several studentstold her to start with 2 � 4. Kerri asked, “Could we start with the 2 � 1?”Some said yes, some no. Kerri smiled and continued the discussion: “OK, tellme what to do.”

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The most confident voices led, using the standard algorithm they hadpreviously learned. As Kerri recorded each step, she pointed out connections tothe work students had been doing with the blocks. Because the speakers werethe most “advanced” students, everyone seemed happy with their answer of 28.

Kerri smiled and then looked puzzled. She reminded the class that somestudents had said it was okay to start with 2 � 1 and asked one student toshow the class how to do that. The student said, “Multiply 2 � 1, and writethe product 2 under the 1.” Kerri pushed him to explain why the 2 needed togo under the 1. He said he wasn’t sure but that he knew the final answerwould be 28. There were several tries at justifying, but no one pointed outthat in this problem, the 2 really represents 20.

Another student, Rachel, suggested writing the 2 under the 1 and puttinga 0 above the 1 in 14. “Finally, the value of 20,” I thought, but nobodypointed this out. Students didn’t like Rachel’s method, saying it would get toocomplicated if the product of the ones were greater than 9. A student namedMegan said she’d tried the strategy, shown in Figure 5.3, which she called the“backwards” method, with 2 � 18.

The class period was over. No one had produced a satisfactory explanationfor why Megan’s “backwards” method worked, but all the students had beenthinking, had strengthened their sense of place value, and were workingtoward an efficient algorithm that matched their understanding. This had

been a difficult discussion forsome, illuminating for others.Allowing students to sharetheir emerging or unconven-tional methods for computa-tion stretches every mind inthe classroom—including theteacher’s! Class ended withthe homework assignment: ashort set of multiplicationproblems to be illustratedwith drawings of base-10blocks and with arithmetic.Some students asked if theycould use the “backwards”method for the latter; Kerri

FIGURE 5.3Megan’s “Backwards”Method of Multiplicationfor 2 � 18

• Multiply 2 x 1 and write down the product, 2,under the 1.

• Multiply 2 x 8 = 16. Write the 6 in 16 under the 8 and 2.

• Write the 1 in 16 above the 1 in 18.• Add the 1 to the 26, arriving at the correct prod-

uct, 36.1

18�

26�

36

1

1

2

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agreed and chose to start the lesson the next day by pushing on student under-standing of place value.

Discourse and Problem Solving

We’ve all seen the same rules of thumb for solving mathematics problems:Make a model or diagram. Make a table, chart, or list. Guess and check. Con-sider a simpler case. Look for patterns. In classrooms in which rich discourseis encouraged, these methods surface within the context of daily lessons.When students solve problems by whatever method makes sense to them andshare their work, many strategies can arise within the context of one problem.Take, for example, the pizza problem alluded to in Chapter 3 (Lappan, 1998):

A large pizza was stored in the cafeteria refrigerator for five nights. Duringthe first night a pizza thief took half of the pizza. The second night thethief took half of what was left. Each night the culprit stole half of theremaining pizza. How much of the pizza was left after the fifth night? Ifthe pizza were left in the refrigerator for a long time, how many nightswould pass before the pizza was completely gone? (p. 46)

Students may use several different strategies to solve this problem. Somemight make a diagram of the pizza, sectioning off the parts eaten each night;others may use arithmetic, creating a simple table showing the amount ofpizza left after each night; still others may set up a coordinate grid and plotvalues in order to look for patterns and make predictions. Students who areconfused need to consider a simpler case involving only one or two nightsbefore being able to work toward the final solution.

Rich, deep, and argumentative discussions occur when students displaytheir work and present their strategies. As students explain their thinking,others can see connections and the usefulness of different methods. It is ourresponsibility as effective teachers to watch the flow of the lesson so thatstudents not only solve the problem, but also learn about the strengths andweaknesses of different strategies. Summarizing and labeling the strategiesmakes them memorable, as does naming them in honor of the students whocame up with them (e.g., Mary’s Method, Theo’s Theory, Pedro’s Plan).

Discourse and Vocabulary

As one teacher said to me, “It’s vocabulary, vocabulary, vocabulary.” For him,the key to students’ understanding was familiarity with mathematical words

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and their meanings. But the traditional approach of handing students a list ofwords and asking them to look up their definitions doesn’t work. Studentsdon’t really know what definitions mean until they’ve grappled with the con-cepts to which the words are applied. What a great opportunity to encouragediscourse! Understanding develops as students use and hear words in context(Bullock, 1994). I suggest that we use classroom discourse to develop the con-cepts before introducing mathematically specific terms. As the NCTM (2000)notes, “It is important to avoid a premature rush to impose formal mathemat-ical language; students need to develop an appreciation of the need for precisedefinitions and for the communicative power of conventional mathematicalterms by first communicating in their own words” (p. 63).

Teachers need to be very attentive as they encourage students to use stan-dard vocabulary when they talk and write. We need to really listen in order touncover misunderstandings; students are apt to parrot back definitions, thusconcealing their confusion. During lessons in which students first encountera new concept, teachers should encourage them to describe ideas in their ownwords before introducing the specialized terms.

In one 5th grade classroom I observed, the teacher wanted students tograpple with the ideas of perimeter and area without introducing the termsthemselves. She began by having the students trace their feet on grid paper.They explored ways to figure out how many squares were within the tracedregion and how many linear units were in the string they had glued aroundthe tracing. As students shared ideas, they spoke about the number of squaresinside the “footprints” and the number of units along the foot’s “ring” or“rim.” Students continued to use these terms as they developed the conceptsof perimeter and area. Over the next few lessons, the teacher frequently usedthe terms “area” and “perimeter” herself, leading her 5th graders to defini-tions and a natural use of the terms.

Using Concept Maps to Foster Discourse

When my 7th graders were working on a unit about circles and investigatingthe relationships between radius, diameter, circumference, and area, I used amodified version of Frayer’s concept map to help them (1969). Many but notall of my students had become comfortable with the estimation that the cir-cumference of any circle is about three of the diameters and that the area isabout three times the radius squared; a few had started using 3.14 (or π)instead of 3 as the value in their formulas. I gave them blank copies of theconcept map and had them write “circumference” in the central portion.

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Definition

Distance around a circle

Facts

Must beabout a circlenot a polygon

Examples

The outside

Lou’s Map

Nonexamples

Not the inside

Circumference

Definition

Path thatmakes the circle

Facts

Circumf =3.14 x diameter

ExamplesWith radius 3

C = 3.14 x 6C = 18.84

Annekki’s Map

Nonexamples

NotC = 3.14 x radius 2

Circumference


Each student worked individually to fill in the sections labeled Definition,Facts, Examples, and Nonexamples. Figure 5.4 shows responses from two ofthe students.

After students had completed their maps, I asked them to show their part-ners what they had written. This type of pair-share is an effective way to helpstudents sort out misconceptions and practice clear explanations prior to afull class discussion.

As the partners shared their models with the whole class, I recorded their ideas on a large sheet of chart paper. The first definition came from Lou: “Circumference is the distance around a circle.” I asked for more definitions andrecorded several, including Annekki’s: “The path that makes the circle.” The classdecided that Lou’s definition was easy to say and understand but confusingbecause it sounded as though he was referring to the space outside the circlerather than the line. Gradually, consensus evolved around Annekki’s definition.

We moved on to the “Facts” section of the concept map. I started the dis-cussion by requesting statements that did not use any formulas. Even thoughI wanted students to understand the circumference and area formulas, it has

FIGURE 5.4 Two Sample Completed Concept Maps

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been my experience that the more general ideas get pushed aside once a dis-cussion of formulas begins. Lou’s comment that “circumference must be abouta circle, not a polygon” was eye-opening for some. Other students talked aboutthe investigations they’d done with string, reminding classmates that theyneeded a bit more than three “diameter strings” to cover the circle’s circum-ference. Examples and nonexamples popped up as the students talked, and thediscussion segued easily into summarization of ideas about the formulas. Therichness of this lesson was in the discourse. Using the concept map withoutthe sharing would have helped students record their own ideas, but would nothave expanded their thinking.

Murray (2004) suggests using concept maps in a way that incorporates a sig-nificant amount of discourse. Her students created concept maps to be fea-tured in parent-student-teacher portfolio conferences at the end of a trimester.Each student created a concept map for a big idea from the semester’s work onoperations and was required to follow a clear set of guidelines. After teacherconferencing and editing, Murray “divided the class into groups of three so that the students could practice their presentations, each student in turnmaking an oral presentation while the remaining two played the roles of par-ents. . . . We could not have found a better way to practice using the languageof mathematics in a meaningful way. During the actual conferences, studentsmade confident presentations and parents expressed appreciation for havinglearned about mathematical relationships they had previously never reallyunderstood” (p. 75).

Summary

As we change our teaching, it is important to realize that we expect studentsto change with us; they also have new responsibilities. The quick-referencetable in Figure 5.5 summarizes teacher and student roles in classroom dis-course as described by the NCTM (1991).

Students may well be uncomfortable shifting from passive observers toactive learners. How can we help them learn these new skills? In some cases,it may help to explicitly explain that the class will operate in a new way; inothers, it may be better to let the students’ new roles develop gradually, as theteacher comes to be seen more as an inquirer.

Regardless of how you make this transition, research shows that studentslearn better when they are actively involved in discussing and arguing about


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ideas. As teachers, it is our mission to get them involved, and classroom dis-course does it! Classrooms become exciting learning spaces when teachersestablish a culture of shared ideas and guide students as they discuss theirmathematical understanding. And when students thrive, teachers experiencethe exhilaration of making it happen.

FIGURE 5.5 Teacher and Student Roles in Classroom Discourse

Teacher’s Role Student’s Role

Poses questions and tasks that elicit, Listen to, respond to, and question theengage, and challenge each student’s teacher and one another.thinking.

Listens carefully to students’ ideas. Use a variety of tools to reason, makeconnections, solve problems, andcommunicate.

Asks students to clarify and justify their Initiate problems and questions.ideas orally and in writing.

Decides which of the ideas students bring Make conjectures and present problems.up to pursue in depth.

Decides when and how to attach math Explore examples and counterexamples to notation or language to students’ ideas. investigate conjectures.

Decides when to provide information, Try to convince themselves and one when to clarify an issue, when to model, another of the validity of particular when to lead, and when to let different representations, solutions, conjectures, students struggle with a problem. and answers.

Monitors student participation in Rely on mathematical evidence and discussions and decides when and how to argument to determine validity.encourage each student to participate.

Source: Adapted from information in Professional Standards for Teaching Mathematics, by the National Council ofTeachers of Mathematics, 1991, Reston, VA: Author.

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87

The Greek word metis is not often heard in current conversation, yet it cap-tures a strong sense of the capabilities and competencies that are a major goalof current mathematics education. David Brooks (2000) describes metis as afaculty that the French might equate with savoir-faire or the Germans withgestalt. The Yale anthropologist James Scott (1998) talks about metis in thecontext of a toolbox of knowledge and practical skills that enables people torespond to change.

One of the primary attributes of metis is that it cannot be taught or mem-orized; it can only be imparted and acquired. Brooks provides the example of the apprentice who may exhaustively learn the rules of cooking but whowill not have the same awareness as a master chef in regard to when the rulesor recipe should be applied and when they should be adjusted. Similarly, abeginning teacher may studiously digest a book on pedagogy, but only ametis-rich teacher will be able to successfully guide students as they struggleto achieve understanding of complex content. Skillful arithmetic studentsmay perform superbly as human calculators, but they will always be surpassedby the metis-endowed mathematics students who have a “feel” for the con-nections and underlying concepts involved in the problem at hand.

I have found that people in metis-rich environments tend to converserather than lecture. They develop a sense of the processes and relationships inthe world around them, and comprehend the complex patterns of learning.Having metis means being aware of the flow of ideas; it means knowingwhich thoughts can go together, and which will never connect. It meansknowing how to react when the unexpected occurs, and how to tell what isreally important from that which is mere distraction. Metis is not scientificknowledge, but rather a special sensitivity to one’s particular circumstances.

Creating Mathematical MetisJoan M. Kenney

6

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People who prize metis welcome a diversity of approaches; people whoacquire it tend to learn by doing, rather than solely by abstract reasoning.

In a series of professional development workshops designed around thetheme of creating and cultivating mathematical metis, I asked participants toproduce a list of attributes that they saw as contributing to a general sense ofmetis. Their lists included such traits as openness, curiosity, resourcefulness,persistence, risk-taking, intuitiveness, and awareness. Participants felt stronglythat if these features are operative in a mathematics classroom, the potential fortrue mathematical metis exists, as evidenced by the seeing and making of con-nections and the ability to generalize to conclusions.

The strategies and examples offered in this book all focus on how to createa sense of comfort with mathematical objects and actions. When this con-nectedness is reinforced, while at the same time the distinction between con-tent and process is kept clear, students are able to move beyond blindmemorization and black-box algorithms. They become skilled at

• Selectively reading mathematics text, • Writing with clarity about their mathematics thinking, • Using a wide range of graphic representations to explore and explain

their mathematics comprehension, and • Discussing their strategies with other students.

Creating Metis for Teachers and Administrators

As important as it is to create metis for students, it is equally vital for teach-ers and administrators to achieve a sense of intuitive mathematical under-standing. They need to collect information on the following issues:

• How does mathematics learning takes place?• What are the important mathematics concepts—the “big ideas”?• What benchmarks can be used to evaluate mathematics teaching? • What are the effects of high-stakes testing on metis in general, and on

mathematics learning in particular?

Current brain research provides us with many insights into how we firstlearn mathematics. New technologies produce a physical mapping of brainactivity that enables us to describe, with far greater precision and sophistica-tion than ever before, how learning takes place. Researchers such as LawrenceLowery (1998) give us new ways to think about what level of mathematics isdevelopmentally appropriate to teach at certain grades. For example, at what

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89Creating Mathematical Metis

point is the human brain ready to make the transition from a two-dimen-sional to a three-dimensional mapping of space? Obviously, such questionshave significant implications for when certain mathematics concepts shouldenter the curriculum. Equally obviously, if mathematics concepts are intro-duced too early, students will suffer an immediate loss of metis. Grade-levelexpectations have been definitively set out in the NCTM standards, and aremirrored to varying degrees by state and district standards. But elementaryand middle school teachers are confronted with a great deal of new curri-culum that has “trickled down” from the upper grades, particularly in theareas of geometry and data and statistics, and these new topics are compet-ing for precious instructional time with an increased emphasis on testing.Therefore, in order to preserve any sense of metis for both students and teach-ers, it becomes vital that the truly important and developmentally appro-priate mathematics concepts be clearly identified at each grade level, and thatthe mathematics be taught through a series of connections rather than as iso-lated lessons.

In their role as instructional leaders, administrators need to have a rigorous,intuitive understanding of mathematics concepts. The expansion of their math-ematical comfort zone will enable them to validate benchmarks and guide pro-grams to successful conclusions; their metis will help them to have an instinctfor when to hold the course, and when it is prudent to change direction.

It is critical that educators document the effect of accountability measures andhigh-stakes tests on mathematics learning. Holding all students to high expecta-tions and agreed-upon standards is a positive goal, but many of the methodsused to assess progress are destructive. If our goal is to give students an aware-ness of connections and a sense of comfort with the mathematical world, theprocess will not be enhanced by curriculum that is devoted to preparing themto answer multiple-choice or single-fact questions. This is the antithesis of metis.

Action Research as an Aid to Metis

Translating education research into practice is a complex and multi-layeredprocess. A study by Hemsley-Brown and Sharp (2003) suggests that actionresearch can stimulate involvement by educators, and that the transfer ofresearch to practice requires strong relationships between researchers andpractitioners. Because the coauthors and researchers of this book are practi-tioners, the story of how the book came about serves to illustrate the mergingof research and practice, and how this merging promotes metis.

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I had been asked to provide professional development for four seniorcoaches from the Massachusetts Mathematics Coaching Program, funded bythe Noyce Foundation. My goal was to design a professional developmentexperience that went beyond the usual model of a scripted workshop. I hadrecently read two books that I found engaging: Barton and Heidema’s TeachingReading in Mathematics (2000), which led me to think about producing a guide-book specifically for the mathematics classroom that would extend the inves-tigation of literacy beyond the topic of reading; and Michael Grady’s Qualitativeand Action Research (1998), a practitioner’s guide to designing research projects,collecting and analyzing data, and reporting research results.

Setting the talented senior coaches to the task of investigating the connec-tions between reading, writing, and other forms of communication in themathematics classroom promised to be an enriching, metis-building endeavor.Practitioners, no matter how experienced, do not tend to see themselves asresearchers or writers; they do not have ready access to many research journals,and are often intimidated by the jargon that clogs so much educational writ-ing. However, if they are provided with a scaffolded, structured environmentin which to assess the research, they can quickly become empowered and stim-ulated to transfer the research to their classroom practice, resulting in the cre-ation of a strong climate of metis for both teacher and student.

In the Appendix to this book you will find agendas for the professionaldevelopment sessions that provided the structured environment in which thisbook was created. Through observing how they are designed and becomingfamiliar with the readings, teachers and administrators will hopefully be stim-ulated to learn more about the language of mathematics, to incorporate thisknowledge into practice, and to create a metis-rich environment for mathe-matics teaching and learning.

Task Clusters

Teachers and administrators can only achieve metis if they have a firmgrounding in the important mathematics concepts, and an idea of how theseconcepts develop across grade levels. To facilitate this process I have found ituseful, as a professional development practice, to create collections of “clus-ter tasks”—groups of four or five mathematics tasks that share a commoncontent topic, such as area and perimeter, estimation, or inverse operations(Kenney, 1999). Each cluster includes a problem from primary, elementary,middle, and high school grades, and the following set of focus questions isprovided for participants to discuss:

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• What is the core mathematics that these tasks have in common?• What additional mathematics concepts or skills are involved?• Are there concepts or skills that are unique to certain tasks?• For what grade level or levels does each task seem most appropriate?

Working the tasks and discussing the focus questions, both in small- andlarge-group settings, provides a unique opportunity to probe for connectionsand have a substantive dialogue about mathematics content. It is interestingthat, at first glance, teachers often feel that certain tasks are too difficult forthe grade level for which they were designed. However, it often turns out thatit is the reading level, not the mathematics, that is perceived as being too dif-ficult. The strategies for improving the reading of mathematics text providedin this book can alleviate this situation.

These task clusters were originally designed to give elementary teachers asense of how the fundamental mathematics concepts they teach will emergeand develop over the ensuing grades; however, the strategy has proved to beequally effective for use in professional development with middle and highschool teachers. Because this group rarely has either the opportunity or theinclination to think about what a basic concept might look like at the kinder-garten or elementary level, the exercise of solving and discussing the clusterof tasks becomes most illuminating. Not only does the point at which con-cepts are refined and extended become clear, but the discussions across gradelevel provide important clues about the misconceptions that occur in theearly years and that can lead to relentless confusion in the upper grades. Forexample, is the fact that students in 9th or 10th grade are so often unable toclearly and mindfully articulate the difference between area and perimeterdue to a lack of understanding that occurs at a certain point in mathematicallearning, or is it something that is inexplicably engrained from the first timethe two concepts are encountered? Or, why do older students immediatelyrush to use an algorithm to solve a problem, but are so often unable to decidewhich formula to use?

What Students Need to Know

Some interesting information has emerged from the Mathematical Associa-tion of America’s (MAA) revision of its recommendations for the undergradu-ate program in mathematics. The MAA Committee on the UndergraduateProgram in Mathematics (CUPM) is primarily concerned with mathematicsinstruction in the first two years of college, and with what incoming students

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need to bring with them from their secondary mathematics experience. Atthe MAA’s annual meeting in 2001, CUPM hosted 13 focus groups involvinga total of over 180 mathematicians; the informal reports of each of thesegroups reveals some strong recurring themes. Here are a few of the character-istics that focus group attendees want to see in their students:

• A comfort, or metis, with symbols and graphs and an ability to usegraphs as a language. Mathematics professors want students who can bothcreate new graphs from given information and analyze provided graphs asdescriptors of mathematical activity. It is thus essential that students beexposed to graphs beginning in the early grades.

• A deep conceptual understanding of their mathematical studies, par-ticularly in regard to functions. One professor rather plaintively wrote that he wished students would “listen to the equations.” Certainly, an ability to betruly “tuned in” to what mathematics symbols are saying is a high form ofmetis.

• The ability to speak the language of spreadsheets. Creating spreadsheetsis a comprehensive mathematical activity; it requires students to fully integratetheir modeling, transforming, inferring, and communicating skills. What asplendid activity to use for assessing mathematical proficiency!

• The ability to show multiple representations of a mathematical problem.One positive outgrowth of the NCTM standards and the use of constructed-response assessments is that young students are being encouraged to visual-ize the mathematical world in a variety of ways, through programs such as“Read It, Draw It, Solve It” (Miller, 2001). With minimal adaptation, theseresources can be used to lead learners to read the initial problem, to draw apicture of what the text is saying, and to represent their solution pictorially,numerically, and verbally. If this becomes a consistent method of problemsolving in elementary school, students develop habits of mind that will standthem in good stead as they move on to the more complex mathematics ofmiddle school, high school, and college.

If students can write clearly about mathematics in both words and in sym-bols, provide graphic representations (either as diagrams or as graphs), andarticulately justify their strategies and how their solutions connect with thoseof their classmates, then they have certainly developed a deep, rich under-standing of the underlying concept.

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Final Thoughts

This book contains a wealth of strategies and suggestions for improving stu-dents’ ability to read mathematics text, to write about their mathematicalthinking, and to enhance their communication through graphic representa-tion and discourse. Some of these suggestions will resonate with you andmatch your teaching style; others may seem cumbersome or time-consuming.To improve your own skills, consider trying one of the strategies that you arenot naturally drawn to. It is only in this way that you can experience firsthandthe discomfort, the cognitive dissonance, and the disruption of metis that atask presented in a certain format, or with a certain unique vocabulary, mayproduce in a student. It is only through this type of experience that webecome sensitive and responsive to all our students, not just to those who seethe mathematical world through the same lens that we do.

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95

Note: Full citations for all readings listed in this appendix can be found inthe References and Resources section.

Session 1

• Goals for the project.• Descriptions of mathematical literacy.• Reflection on how each participant acquired a second language (open

discussion).• Discussion of common threads in language acquisition.• Beginning list of mathematics vocabulary.• Working with visual representations

– That stimulate vocabulary development.– That inform text.

Assignment: Create a concept map for “percent” and a Frayer model for“prime.”

Reading: “Using Graphic Organizers to Improve the Reading ofMathematics” (Braselton & Decker, 1994)

Session 2

• Presentation of concept maps and Frayer models.• Reflection on how each participant acquired mathematical language

(open discussion).• Discussion of “Teaching the Language of Mathematics” (Krussel, 1998).• Refining of graphic organizers for specialized use in mathematics.

Appendix

Structured Agendas Used to Research This Book

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Assignment: Write a comparison of personal experience in acquiring asecond language, and in acquiring mathematical language.

Readings: • Semantic Aspects of Quantity (Schwartz, 1996)• Assessing Mathematical Skills and Understanding Effectively (Schwartz &

Kenney, 1999)• “Handwriting Mastery” (Allen, 2003)• Text Organization and Its Relation to Reading Comprehension (Dickinson,

Simmons, & Kameenui, 2000)

Session 3

• Sharing of language acquisition comparisons.• Sharing of reading assignments from last meeting.• Introduction to Reading and Writing in Mathematics, 2nd ed. (Barton &

Heidema, 2002).• Introduction to action research and Qualitative & Action Research (Grady,

1998).• Exploration of various bibliographies using a library research catalogue.• Document search at university library.Assignment: Write up a draft focus paper.Readings:• “Strategies to Support the Learning of the Language of Mathematics”

(Rubenstein, 1996)• “Mathematics as a Language” (Usiskin, 1996)• “Using Student Contributions and Multiple Representations to Develop

Mathematical Language” (Herbel-Eisenmann, 2002)

Session 4

Language work update and overview:• Question: “Why is it so difficult?”• Discussion of personal experiences of language impeding mathematical

learning.• Discussion of content development and literacy issues using sample

tasks from Balanced Mathematics Assessment for the 21st Century (Schwartz,2000).

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97Appendix

• Discussion of focus questions:– What difficulties does each task represent in the areas of vocabu-

lary, format, reading level, lack of clarity, and ambiguousness?– What changes would you make to the text in order to facilitate

access to the mathematics?

Session 5

Continuation of literature review: • “The Role of Reading Instruction in Mathematics” (Curry, 2004)• “Teaching Content Area Vocabulary” (Graves, 2004)• “Literacy in the Language of Mathematics” (Bullock, 1994)Assignment: Write up an incident around mathematical language that

you observe in a classroom, or interview a teacher regarding language issues.Readings: The following articles are all from the November 2002 issue

of Educational Leadership:• “From Efficient Decoders to Strategic Read” (Vacca)• “Teaching Reading in Mathemathics and Science” (Barton, Heidema, &

Jordan)• “Advanced Math? Write!” (Brandenbury)• “Seven Literacy Strategies That Work” (Fisher, Frey, & Williams)

Session 6

• Sharing of draft chapter summaries.• Continuation of literature review:

– What Mathematical Knowledge Is Needed for Teaching Mathematics?(Ball, 2003)

– “Language Pitfalls and Pathways to Mathematics” (Barnett-Clarke& Ramirez, 2004)

– “The Role of Reading Instruction in Mathematics” (Curry, 2004)– “Math Lingo vs. Plain English: Double Entendre” (Hersh, 1997)

Session 7

Discussion of questions raised at Boston Higher Education Conference onHigh School Literacy, April 2003:

• What does it mean to be literate in different content areas?• What do we mean when we say that students can’t read?

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• What do students need to do to read for meaning in a content class-room?

• What is the content teacher’s responsibility for teaching students to usereading as a learning tool?

Readings: • Writing to Learn Mathematics: Strategies That Work, K–12 (Countryman,

1992)• Writing in Math Class: A Resource Guide for Grades 2–8 (Burns, 2002)

Session 8

Report on progress of writing, difficulties, and changes in direction.Reading: “Reporting Research Results” (Grady, 1998, pp. 35–42)

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99

Note: Included in the following bibliography are full citations for all works explicitlycited within the body and appendix of this book, as well as additional resources thatwe found valuable and significant in our research.

Albert, L. R. (2000). Out-in—inside-out: 7th grade students’ mathematical thoughtprocesses. Educational Studies in Mathematics, 41(2), 109–141.

Albert, L. R., & Antos, J. (2000). Daily journals connect mathematics to real life.Mathematics Teaching in the Middle School, 5(8), 526–531.

Allen, J. (2003). Yellow brick roads: Shared and guided paths to independent reading.Portland, ME: Stenhouse Publishers.

Allen, R. (2003, Summer). Handwriting mastery: Fluent form is crucial for expres-sion. ASCD Update.

Anderson, N. C. (2002, November). Student explanations of mathematical reasoning.Paper presented at the National Council of Teachers of Mathematics EasternRegional Conference, Boston, Massachusetts.

Armstrong, T. (2003). The multiple intelligences of reading and writing. Alexandria, VA:Association for Supervision and Curriculum Development.

Aspinwall, L., & Aspinwall, J. (2000). Investigating mathematical thinking usingopen writing prompts. Mathematics Teaching in the Middle School, 8(7), 350–353.

Association of Teachers of Mathematics in Maine. (2000, Winter). I have, who has?ATOMIM Newsletter, 11.

Baldwin, R., Ford, J., Smith, C. (1981). Teaching word connotations: An alternativestrategy. Reading World, 21, 103–108.

Ball, D. L. (2003). What mathematical knowledge is needed for teaching mathematics?Remarks prepared for Secretary’s Summit on Mathematics, U.S. Department ofEducation.

References and Resources

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Barnett-Clarke, C., & Ramirez, A. (2004). Language pitfalls and pathways to mathe-matics. In R. Rubenstein (Ed.), Perspectives on the teaching of mathematics. Reston,VA: National Council of Teachers of Mathematics.

Barton, M. L., & Heidema, C. (2000). Teaching reading in mathematics. Aurora, CO:Mid-Continent Research for Education and Learning.

Barton, M. L., & Heidema, C. (2002). Teaching reading in mathematics (2nd ed.).Aurora, CO: Mid-Continent Research for Education and Learning.

Barton, M. L., Heidema, C., & Jordan, D. (2002, November). Teaching reading inmathematics and science. Educational Leadership, 60(3), 24–28.

Baxter, J., Woodward, J., Olson, D., & Robyns, J. (2002). Blueprint for writing in middle school mathematics. Mathematics Teaching in the Middle School, 8(1), 52–56.

Behrmann, M. M. (1995). Assistive technology for students with mild disabilities.(ERIC Digest E529)

Berger, A. (1989). Ways of activating prior knowledge for content area reading. In D. Lapp, J. Flood, & N. Farnan (Eds.), Content area reading and learning:Instructional strategies (pp. 118–121). Edgewood Cliffs, NJ: Prentice-Hall.

Borasi, R., Siegel, M., Fonzi, J., & Smith, C. (1998). Using transactional readingstrategies to support sense-making and discussion in mathematics classrooms.Journal for Research in Mathematics Education, 29(3), 275–305.

Brandenbury, M. L. (2002, November). Advanced math? Write! EducationalLeadership, 60(3), 67–68.

Branford, J. D., Brown, A., & Cocking, R., (Eds.). (2000). How people learn: Brain,mind, experience, and school. Washington, DC: National Research Council.

Braselton, S., & Decker, B. (1994). Using graphic organizers to improve the reading ofmathematics. The Reading Teacher, 48(3), 276–281.

Brooks, D. (2000). Bobos in paradise. New York: Simon & Schuster.Bullock, J. O. (1994, October). Literacy in the language of mathematics. The American

Mathematical Monthly, 101(8), 735–743.Burns, M. (2002). Writing in math class: A resource guide for Grades 2–8. Sausalito, CA:

Math Solutions Publications.California State University Writing Center. (2003). Helping ESL students improve their

writing. Los Angeles, CA: Author.Carroll, J. A. (1991). Drawing into meaning: A powerful writing tool. English Journal,

80(6), 34–38.Chamot, A., & O’Malley, J. (1994). The CALLA handbook: How to implement the

Cognitive Academic Language Learning Approach. Reading, MA: Addison-Wesley.Clement, L. (2004). A model for understanding, using, and connecting representa-

tions. Teaching Children Mathematics, 11(5), 97–102.Countryman, J. (1992). Writing to learn mathematics: Strategies that work, K–12.

Portsmouth, NH: Heineman.


101References and Resources

Curry, J. (2004). The role of reading instruction in mathematics. In D. Lapp, J. Flood,& N. Farnan (Eds.), Content area reading and learning (2nd ed., pp. 227–241).Mahwah, NJ: Lawrence Erlbaum.

Dehaene, S. (1998). The number sense: How the mind creates mathematics. New York:Oxford University Press.

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105

abstractionmodeling for, 70of numbers, 3–4

action research, 89–90active learning, 85–86addition

in curriculum, 3of fractions, 53–54, 54fas stand-in for multiplication, 45

administrators, role of, in mathematicsinstruction, 88–89

algebraconceptual capability in, vs. arithmetic,

46in secondary education, 3

algorithms, in curriculum, 3anchor papers, 32angles, mathematical vs. other, 39answers. See responsesarea (geometric)

different understandings of, 35–36vs. perimeter, 91

arguments, value of, 78arithmetic

discourse and, 80–82inadequacy at, vs. conceptual capabilities,

46articles (grammatical)

ESL students and, 39in technical syntax, 51

articulation, importance of, 70assessment. See also tests

anchor papers and, 32multiple-choice tests and, 33NCTM on, 32open-ended questions and, 32–33, 33fof problem solving, 2self, CALLA strategy for, 40ftechniques for, 32writing and, 31–33

Balanced Assessment Program, 2Barton, Mary Lee, 90“base,” different meanings of, 51breaking even, 57–59Bullock, James, 53

calculus, conceptual capability at, vs.arithmetic, 46

California State University Writing Center, 39CALLA, 39–44, 40f, 41f, 42fcircumference, concept mapping and, 83–85,

84fCognitive Academic Language Learning

Approach. See CALLAcollege level, 91–92communicating

definition of, 2with show and tell, 61

concept maps, 83–85, 84f

Index

Note: Information presented in figures is denoted by f.

09--Index--105-110 9/23/05 9:12 AM Page 105


concrete representations, deficits in, 48confusion

with concept of “more,” 66–69with decimals, 65with graphic devices, 56–57with vocabulary, 58–59

confusion, benefits from, 78conjunctions, in technical syntax, 51Connected Mathematics Project, 33–34content, vs. process, 1content dimension, 1context

mathematical vs. conversational, 68in word problems, 67

curriculum, 3

data, as mathematical nouns, 2decimals, confusion over, 65definitions

special, of familiar words, 51student-derived, 35–36

discourse, 72–86benefits of, 73, 74computation and, 80–82concept maps and, 83–85, 84fdefinition of, 72favored activities and, 76fostering, 74–80furniture arrangement for, 75incorrect ideas in, 78leadership and, 79management of, 76NCTM on, 72open-ended questions and, 78peer review and, 79–80probing, 73problem solving and, 82questions for, 79respect in, 78rich, 73student roles in, 86fteacher roles in, 86ftraditional, 73vocabulary and, 82–83

divisionin curriculum, 3decimals in, 65–66

domain specificity, of mathematical skills, 1double meanings, avoiding, 69

DragonDictate, 48Dragon Naturally Speaking Software, 48drawing. See also graphic devices; modeling;

representations; visual learningfor abstraction of concepts, 70benefits of, 69, 71geometric learners and, 71kinesthetic learners and, 71materials for, 70open-ended questions and, 70providing time for, 70slowing effect of, 71

education, goals of, 74elementary school

arithmetic curriculum in, 3show and tell in, 61trickle down topics in, 89writing in, 32

embarrassment, of students, overmisunderstandings, 69–70. See also ridicule

endings, word, 70enunciation, importance of, 70ESL students, 38–44

CALLA and, 39–44, 40f, 41f, 42fgrammar problems with, 39simplicity of language and, 39

“even,” 57–59exponential decay, 60–61

factors (mathematical), special-needs studentsand, 45

flow charts, for writing, 49foreign language. See ESL studentsformal vs. informal mathematics, 3formulating. See modelingfraction(s)

addition of, 53–54, 54farea models for, 63–65, 64fdecimals and, 65logic and, 65–66, 66fprobability notation with, 59–60, 59fstrips, 62–63, 63fvisualization of, 55, 62–63, 63f

functionsimportance of understanding, 92as mathematical nouns, 2modeling of, 2

furniture, arranging, to foster discourse, 75

09--Index--105-110 9/23/05 9:12 AM Page 106

107Index

games, to foster discourse, 76, 77fgeometric learners, 71geometric objects, communication and, 2goals, of education, 74grade-level expectations, 89Grady, Michael, 90grammar, nouns and verbs in, 2graphic devices. See also drawing; modeling;

visual learningfor addition of fractions, 53–54, 54fconfusion arising from, 56–57line graphs as, 42–44, 42f, 57–59oral interpretation of, 62organizers, 70–71student drawings as, 69writing and, 29–30

groups, working inwith concept maps, 83–85, 84findividual differences and, 29–30ridicule and, 30writing and, 27–28, 28f, 29f, 30

Harvard Graduate School of Education, 2Heidema, Clare, 90history, popular concepts in, of education, vHood Children’s Literacy Project, 39hypotenuse, as technical term, 51

icons, 70“I Have, Who Has?” game, 76, 77finferring, definition of, 2infinity, discourse exploring concept of, 74–75Inspiration (software), 46integer, as technical term, 51

jokes, embarrassment over misunderstandingas, 69–70

journal keeping, 31, 32justification, of responses, 31, 50

kinesthetic learning, 71

Lamour, Nekita, 39language(s)

commonality in, 1confusing aspects of mathematical, 51learning of, vs. mathematics, 3–4native, 39pictorial, mathematics as, 52

visualization of, 70–71world, 1

leadership, 79learning styles. See active learning; geometric

learners; kinesthetic learning; visuallearning

lesson planning, order of representationalselection and, 70

location, of teacher, 76locutions, 58logic

vs. domain specific skills, 1problems with, 65–66, 66f

logograms, 51Lowery, Lawrence, 88

manipulatingdefinition of, 2vs. modeling, 2as skill-based action, 2

Massachusetts Mathematics CoachingProgram, 90

Mathematical Association of America, 91–92mathematics

computers and need for, 44conceptual level of, 50connections within, 70as foreign language, 3formal vs. informal, 3language learning vs., 3–4language of, confusing aspects in, 51as pictorial language, 52syntax of, 52visualization of language of, 70–71writing and, need for, 50

MATHPLAN program, 45, 46f, 47fmeasurements, as mathematical nouns, 2memory, special-needs students and, 45, 48metaphor, monitoring use of, 69metis, 87–92

acquisition of, 87action research and, 89–90for administrators, 88–89definition of, 87for teachers, 88–89traits for, 87–88

modeling. See also graphic devices;representations

actual vs. representational, 54

09--Index--105-110 9/23/05 9:12 AM Page 107


modeling—continueddefinition of, 2in different dimensions, 53–54fractions, 53–55, 54f, 62–63, 63–65, 63f, 64fgraphical, 53–54, 54fvs. manipulation, 2personal experience and, 55–59, 56f, 58fprobability, using fractions, 59–60, 59fuse of, for move to abstraction, 70

“more,” confusion over concept of, 66–69multiple-choice tests, 33, 89multiplicationaddition as stand-in for, 45in curriculum, 3different methods for, 80–81, 81fplace-value in, 81–82Murray, Miki, 76, 85

NCTM (National Council of Teachers ofMathematics)

on assessment, 32on discourse, 72, 85, 86fon mathematical language, 38standards-based curriculum and, von vocabulary, 83on writing, 49

nounscount, 39ESL students and, 39in grammar, 2in languages, 1mathematical, 2noncount, 39in word problems, changing meaning of,

68Noyce Foundation, 90numbers

in abstract, 3–4communication of, as objects, 2as mathematical nouns, 2

objects, mathematicalcommunication of, 2curriculum and, 3

“or,” conversational vs. mathematical, 51organization

with graphic organizers, 70–71software for, 45, 48special-needs students and, 45

outliningcomputer-assisted, 48–49for writing, 49

paraphrasing, pitfalls of, 75patterns, as mathematical nouns, 2peer pressure, question asking and, 69–70peer review, of ideas, 79–80peers, listening to, vs. teacher, 76perimeter, 73–74, 73f, 91pictorial language, mathematics as, 52Pisauro, Jaqueline, 79place-value, in multiplication, 81–82positive intent, in student speech, 69–70prepositions

ESL students and, 39in technical syntax, 51

Principles and Standards for School Mathematics(NCTM), 72

prior knowledge, 53probability

fractional notation in, 59–60, 59fprobability, conceptual capability at, vs.

arithmetic, 45problem solving

actions for, 2assessment of, 2CALLA strategies for, 39–44, 40f, 41fdevelopment of math literacy with, 39discourse and, 82open-ended response questions and,

32–33, 33f, 34f, 35f, 36f, 37fsharing of methods for, between students,

78writing and, 27–30

process(es)vs. content, 1describing, ESL students and, 39

process dimension, 1pronouns

referents, 70in technical syntax, 51

Qualitative and Action Research (Grady), 90quantity, 67–69questions

as answers, 80open-ended, 32–33, 33f, 34f, 35f, 36f, 37f,

70, 78

09--Index--105-110 9/23/05 9:12 AM Page 108

109Index

positive intent in, 69–70student to student, 75, 78

real-life experience, 39reasoning

actions in, 2unconventional, of students, 76

representations. See also modeling; visuallearning

concrete, deficits in, 46translating between, 70

research, action, 89–90respect, of wrong answers, 78responses

justification of, 31, 50listening to, 78wrong, respect of, 78

revision, benefits of, in writing, 30–31rhombus, as technical term, 51ridicule. See also embarrassment

in group work, 30

Scott, James, 87secondary education, algebra in, 3shapes, as mathematical nouns, 2shopping, mathematics and, 44show and tell, 61slope, 56–57software

pitfalls of, 49voice-activation, 48for writing organization, 48, 49

spaces, as mathematical nouns, 2special-needs students

addition and, 45factors and, 45memory and, 45multiplication and, 45organizational skills of, 48software for, 48, 49technology and, 45writing and, 44–48

spreadsheets, 92student-derived definitions, 35–36students, ESL. See ESL studentsstudents, special needs. See special-needs

studentssubtraction, in curriculum, 3symbols, 51. See also graphic devices

syntax, 61–69mathematical, 52nuance in, 62technical, 51

tangents, conversational, value of, 78task clusters, 90–91Teaching Mathematics Vocabulary in Context

(Murray), 76Teaching Reading in Mathematics (Barton &

Heidema), 90technical symbols, 51technology

pitfalls of, 49ubiquity of, 44–45

Technology-Enhanced Learning EnvironmentWeb site, 49

tense, ESL students and, 39terminology. See vocabularytests. See also assessment

graphic organizers in preparation for, 71multiple-choice, 33, 89open-ended questions as preparation for,

32textbooks

copying pages from, to aid engagement, 70graphic distractions in, 1page format in, 1

transforming, definition of, 2

unconventional reasoning, of students, 76

Verbal and Visual Word Association, 71verbs

in grammar, 2in languages, 1mathematical, 2

ViaVoice, 48visual learning. See also drawing; graphic

devices; representationsaccommodating, 71example of, 4exponential decay and, 60–61

vocabularyconfusion over, 58–59definitions and, 35–36discourse and, 82–83ESL students and, 38–39of exponents, 60–61

09--Index--105-110 9/23/05 9:12 AM Page 109


vocabulary—continuedimproving, 57NCTM on, 83standard, 83teacher, monitoring, 69technical, 51use of student, 75writing and, 33–34

voice-activation software, 48Voice Express, 48

webbing, for writing, 49Web sites

for writing help, 49word endings, 70word problems

CALLA strategy for, 40fcontext in, 67extraneous information in, 66–67nouns in, changing meaning of, 68pitfalls of, 66–69

writinganchor papers and, 32

assessment of learning with, 31–33, 50benefits of, 31, 50directed expository, 31at elementary level, 32ESL students and, 38–44graphic devices and, 28group work and, 27–30NCTM on, 49need for, in mathematics, 27organization of past projects, 31responses to problems, 27–31, 28f, 29frevision in, 30–31software for organization of, 46, 48special needs students and, 44–46structured, 38, 38f, 46, 47f, 48fstudent-teacher communication and, 50supports, 48–49teachers and, 28vocabulary and, 33–34voice-activation software and, 48Web site for help with, 49

09--Index--105-110 9/23/05 9:12 AM Page 110

111

Joan M. Kenney’s professional career has encompassed a wide variety of expe-riences in the field of mathematics. She has worked as a research scientist, spe-cializing in operations analysis and risk management; taught mathematics atthe secondary and college levels; and performed task modeling and pedagogi-cal intervention in elementary and middle school classrooms. Joan served asthe national evaluator for the National Science Foundation’s Assessment Com-munity of Teachers and Connecting with Mathematics projects, the Councilfor Basic Education’s Instructional Leadership Academy, and the Digi-Blockprogram. She has delivered keynote addresses at several national and interna-tional conferences, and has written extensively about mathematics educationreform and assessment.

Joan recently retired from the Harvard Graduate School of Education,where for 10 years she was the Project Coordinator and Codirector of the Bal-anced Assessment Program. During that time she was involved in assessmenttask design, student performance evaluation, and outreach to communitystakeholders; she also served on the Mathematics Task Force of the Massachu-setts Board of Higher Education, and on the original design committee for theMassachusetts Comprehensive Assessment System. She continues to consultwith school districts on issues of mathematics curriculum and classroom prac-tice, and to provide professional development for teachers and administratorsin the areas of mathematics content and assessment. Joan may be contactedby e-mail at [email protected].

About the Authors

10--About Authors--111-113 9/23/05 9:13 AM Page 111


Euthecia Hancewicz is a mathematics teacher coach, trainer, and consultantcurrently working with middle and elementary schools. She has taught math-ematics from grades 2 through 10, focusing mainly on middle school students.

As a participant in the Noyce Foundation’s Massachusetts MathematicsCoaching Project, Euthecia witnessed firsthand how teachers worked tochange their practice in light of current research. During those years, shebegan to write about the power of classroom discourse—a valued part of herpersonal teaching style.

Euthecia earned a Bachelor of Arts in Psychology at Westhampton College,University of Richmond, Virginia, and a Masters of Education from the Uni-versity of Massachusetts at Amherst. She is an active member of the NationalCouncil of Teachers of Mathematics (NCTM), the National Council of Super-visors of Mathematics, and the National Staff Development Council, and hasbeen a presenter at the NCTM national conference. She continues the coach-ing and facilitating work to further her conviction: “The best way to helpyoung people learn to think is to help them learn mathematics.” Eutheciamay be reached at [email protected].

Loretta Heuer is a Senior Research Associate at Education Development Cen-ter’s (EDC) K–12 Mathematics Curriculum Center in Newton, Massachusetts. Aformer elementary school teacher, Loretta has worked as a middle schoolmathematics coach and as an implementation advisor in urban and suburbandistricts through the Massachusetts Mathematics Coaching Project. She hastrained elementary school principals in mathematics using EDC’s Lenses onLearning program, and is currently coaching a team in EDC’s Lesson StudyCommunities project. Loretta’s main research interests are the use of inter-active technology in professional development and the role of graphic repre-sentation in the learning of mathematics. She can be reached at [email protected].

Diana Metsisto started her career as a programmer and systems analystworking on scientific, real-time systems. After 11 years she switched to edu-cation, where she worked as a 7th grade mathematics teacher for 23 years inNorwell, Massachusetts. She then served as a mathematics coach for threeyears under the auspices of the Massachusetts Middle School MathematicsProject. For the past two years, Diana has been a consultant for the EducationDevelopment Center, working as a coach with teams of middle and highschool mathematics teachers for the Lesson Study Communities Project.


113About the Authors

Diana received a Bachelor of Arts in Mathematics from Northeastern Univer-sity and a Master of Arts in Critical and Creative Thinking from the Universityof Massachusetts. She is interested in finding ways for students and teachers todevelop mathematical understanding and power and in researching how wethink about and solve problems and develop conceptual understanding. Dianabelieves that this is best accomplished by listening, dialoguing, and reflectingon learning experiences. She may be reached at [email protected].

Cynthia L. Tuttle taught mathematics for 21 years at both the elementaryand middle school levels and worked as a certified reading teacher. She servedas an Assistant Professor at both Castleton State College and American Inter-national College, where she was the first Coordinator of Supportive Servicesfor Undergraduates with Learning Disabilities. She has been a mathematicscoach in middle school classrooms for the Massachusetts Mathematics Coach-ing Project, and has worked with teachers through the Hampshire EducationalCollaborative. At present, she works independently as a mathematics coachand consultant.

Cynthia earned her Doctorate of Education at the University of Massachu-setts. Throughout her career, she has concentrated on strategies to help all stu-dents, including those with learning disabilities, succeed in mathematics. Shebelieves strongly that every student has a right to achieve his or her full poten-tial in mathematics and that every teacher has a responsibility to assist inaccomplishing this goal. Cynthia may be reached at [email protected].


Related ASCD ResourcesLiteracy Strategies for Improving Mathematics Instruction

At the time of publication, the following ASCD resources were available; for themost up-to-date information about ASCD resources, go to www.ascd.org. ASCD stocknumbers are noted in parentheses.

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Improving Mathematics Instruction Through Coaching by Glenda Copeland (#505298)

Reading Strategies for the Math Classroom by Rachel Billmeyer (#205063)

Three Steps to Math and Reading Success by Lauren Armour (#202202)

Books

The Beginning Schools Mathematics Project by Anne McKinnon and Don Miller(#195264)

Math Wonders to Inspire Teachers and Learners by Alfred S. Posamentier (#103010)

Video

The Brain and Mathematics, Tape 1: Making Number Sense (1 tape and facilitator’sguide, #400238)

The Lesson Collection: Math Strategies, Tapes 17–24 (8 tapes, #401044)

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